Cfx Theory Tutorial
description
Transcript of Cfx Theory Tutorial
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33Full Buoyancy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34Boussinesq Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Multicomponent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35Multicomponent Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35Scalar Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35Algebraic Equation for Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37Constraint Equation for Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37Multicomponent Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38Energy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39Multicomponent Energy Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
Table of Contents: Turbulence and Wall Function Theory
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Additional Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40Diffusive Transport Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41Algebraic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
Rotational Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42Alternate Rotation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43Momentum Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43General Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45Mass (Continuity) Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45Bulk Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61Adaption Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61Mesh Refinement Implementation in ANSYS CFX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62Mesh Adaption Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
Flow in Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65Darcy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65Directional Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67
Turbulence and Wall Function Theory
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69Statistical Turbulence Models and the Closure Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70
Eddy Viscosity Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72The Zero Equation Model in ANSYS CFX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74Two Equation 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 for Reynolds 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
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Detached Eddy Simulation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100SST-DES Formulation Strelets et 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Mathematical Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Wall Distance Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171D Illustration of Concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Concept Generalized to 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
GGI and MFR Theory
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Interface Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Multiphase Flow Theory
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Multiphase Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Multiphase Total Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
The Homogeneous and Inhomogeneous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125The Inhomogeneous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125The Homogeneous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Inhomogeneous Hydrodynamic 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 Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Interphase Drag for the Free Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Lift Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Virtual Mass Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Wall Lubrication Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Interphase Turbulent Dispersion Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Solid Particle Collision Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Solids Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Solids Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Solids Bulk Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Solids Shear Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Granular Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Interphase Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Phasic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Inhomogeneous Interphase Heat Transfer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Homogeneous Heat Transfer in Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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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 the Algebraic Slip Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Turbulence Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Energy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Wall Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Turbulence Modeling in Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Phase-Dependent Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Turbulence Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Homogeneous Turbulence for Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Additional Variables in Multiphase Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Additional Variable Interphase Transfer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Homogeneous Additional Variables in Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Sources in Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Fluid-specific Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Bulk Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Secondary Fluxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171User Defined Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172General Species Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 or Added Mass Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Pressure Gradient Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Turbulence in Particle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Turbulent Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
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Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Simple Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Liquid Evaporation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Oil Evaporation/Combustion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Coal Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Hydrocarbon Fuel Analysis Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Basic Erosion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Model of Finnie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Model of Tabakoff and Grant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206Overall Erosion Rate and Erosion Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Spray Breakup Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Primary Breakup/Atomization Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Secondary Breakup Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Dynamic Drag Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Dynamic Drag Law Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Penetration Depth and 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 Flamelet Model for Non Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Coupling of Laminar Flamelet with the Turbulent Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Flamelet Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249Mueller Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Table of Contents: Radiation Theory
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Spark Ignition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Phasic Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
NO Formation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252Formation Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Chemistry Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Soot Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Soot Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Soot Combustion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261Turbulence Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Radiation Theory
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Blackbody Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265Quantities of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Radiation Through Domain Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Rosseland Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Wall Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
The P1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Wall Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 of the Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277The Coupled System of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Solution Strategy - The Coupled Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Linear Equation Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293Residual Normalization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Discretization Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Controlling Error 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 model fluid flow, heat, and mass
transfer in ANSYS CFX for single-phase, single and multi-component flow without
combustion or radiation. It is designed to be a reference for those users who desire a more
detailed understanding of the mathematics underpinning the ANSYS CFX-Solver, and is
therefore not essential reading. It is not an exhaustive text on CFD mathematics; a reference
section is provided should you wish to follow up this chapter in more 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 related subjects:
• Further Background Reading (p. 6 in "ANSYS CFX Introduction")
Basic Solver Capability Theory: Documentation Conventions
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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 the fundamental magnitudes of
length ( ), mass ( ), time ( ), temperature ( ) and chemical amount ( ).
List of Symbols
This section lists symbols used in this chapter, together with their meanings, dimensions
and where applicable, values. Dimensionless quantities are denoted by 1. The values of
physical constants (or their default values) are also given.
More information on the notation used in the multiphase and multicomponent chapters is
available.
• Multiphase Notation (p. 124)
• Multicomponent Notation (p. 35).
L M T Θ 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– Θ 1–
Ch 1
CR1 M L 3– T 1–
CR2 M L 4–
Cε1 k ε 1 1.44
Cε1RNG k ε 1 1.42 f h–
Cε2 k ε 1 1.92
Cε2RNG k ε 1 1.68
Cµ k ε 1 0.09
CµRS 1
CµRNG k ε 1 0.085
c L T 1–
Basic Solver Capability Theory: Documentation Conventions
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concentration of components A and Bi.e. mass per unit volume ofcomponents 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
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– Θ 1–
cv L2 T 2– Θ 1–
cε1 1 1.45
cε2 1 1.9
DAB L2 T 1–
DΦΓΦ ρ⁄ L2 T 1–
d L
E 1 9.793
f µ 1 0.01
f h k ε 1
g L T 2–
h hstat, L2 T 2–
hc M T 3– Θ 1–
htot L2 T 2–
k L2 T 2–
M U c⁄ 1
m 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– Θ 1– 8314.5
Basic Solver Capability Theory: Documentation Conventions
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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, see StaticTemperature (p. 9).
domain temperature For details, seeDomain Temperature(p. 9).
buoyancy reference temperature usedin the Boussinesq approximation
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 theBoussinesq approximation)
RNG - turbulence model constant
diffusivity
molecular diffusion coefficient of
component
Symbol Description Dimensions Value
Re rU d m⁄ 1
r L
rα α 1
SE M L 1– T 3–
SM M L 2– T 2–
SMS M L 3– T 1–
Sct µt/Γt 1
sαβ αβ
M T 1–
T T stat, Θ
Tdom Θ
T ref Θ
T sat Θ
T tot Θ
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
αα
ββ
β Θ 1–
βRNG k ε 1 0.012
Γ M L 1– T 1–
ΓAA M L 1– T 1–
Basic Solver Capability Theory: Documentation Conventions
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Subscripts Quantities which appear with subscripts , , refer to that quantity for component ,
, in a multicomponent fluid.
Quantities which appear with subscripts , , refer to that quantity 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
ΓΦ M L 1– T 1–
Γt M L 1– T 1–
δ 1
ε L2 T 3–
ζ M L 1– T 1–
κ 1 0.41
λ M L T 3– Θ 1–
µ M L 1– T 1–
µt M L 1– T 1–
µeff µ µt+ M L 1– T 1–
ρ M L 3–
Pr cpµ λ⁄ 1
Prt cpµt λt⁄ 1
σk k 1 1.0
σε k ε 1 1.3
σω k ω 1 2
σεRS 1
σkRNG k ε 1 0.7179
σεRNG k ε 1 0.7179
τ M L 1– T 2–
υ M 1– L3
Φ M L 3–
φ
ω T 1–
A B C A
B C
α β γ α β γ
Basic Solver Capability Theory: Documentation Conventions
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Such quantities are only used in the chapters describing multicomponent and multiphase
flows.
• Multicomponent Flow (p. 35)
• Multiphase Flow Theory (p. 123)
Variable Definitions
IsothermalCompressibility
The isothermal compressibility defines the rate of change of the system volume with
pressure. For details, see Variables Relevant for Compressible Flow (p. 62 in "ANSYS
CFX-Solver Manager User's Guide").
(Eqn. 1)
IsentropicCompressibility
Isentropic compressibility is the extent to which a material reduces its volume when it is
subjected to compressive stresses at a constant value of entropy. For details, see Variables
Relevant for Compressible Flow (p. 62 in "ANSYS CFX-Solver Manager User's Guide").
(Eqn. 2)
ReferencePressure
The Reference Pressure (Eqn. 3) is the absolute pressure datum from which all other
pressure values are taken. All relative pressure specifications in ANSYS CFX are relative to the
Reference Pressure. For details, see Setting a Reference Pressure (p. 10 in "ANSYS
CFX-Solver Modeling Guide").
(Eqn. 3)
Static Pressure ANSYS CFX solves for the relative Static Pressure (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 velocity components 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 modified pressure. This variable is
named Pressure in ANSYS CFX.
(Eqn. 6)
1ρ---–⎝ ⎠
⎛ ⎞ dρdp------⎝ ⎠
⎛ ⎞T
1ρ---–⎝ ⎠
⎛ ⎞ dρdp------⎝ ⎠
⎛ ⎞S
Pref
pstat
pabs pstat pref+=
k ε
k
p' pstat2ρk
3---------+=
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Static Enthalpy Specific static enthalpy (Eqn. 7) is a measure of the energy contained in a fluid per unit mass.
Static enthalpy is defined in terms of the internal energy of a fluid and the fluid state:
(Eqn. 7)
When you use the thermal energy model, the ANSYS CFX-Solver directly computes the static
enthalpy. General changes in enthalpy are also used by the solver to calculate
thermodynamic properties such as temperature. To compute these quantities, you need to
know how enthalpy varies with changes in both temperature and pressure. 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. For most materials the first
term always has an effect on enthalpy, and, in some cases, the second term drops out or is
not included. For example, the second term is zero for materials which 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 heat when you run the thermal
energy equation model.
Material with Variable Density and Specific HeatIn order to support general properties, which are a function of both temperature and
pressure, a table for is generated by integrating Equation 9 using the functions
supplied for and . The enthalpy table is constructed between the upper and lower
bounds of temperature and pressure (using flow solver internal defaults or those supplied
by the user). For any general change in conditions from to , the change
in enthalpy, , is calculated in two steps: first at constant pressure, and then at constant
temperature using Equation 10.
(Eqn. 10)
hstat ustatpstatρstat---------+=
dh h∂T∂
------⎝ ⎠⎛ ⎞
pdT h∂
p∂------⎝ ⎠
⎛ ⎞T
dp+=
dh cpdT 1ρ--- 1 T
ρ--- ρ∂
T∂------⎝ ⎠
⎛ ⎞p
+ dp+=
cp ρ
h T p,( )
ρ cp
p1 T1,( ) p2 T2,( )
hd
h2 h1– cpT1
T2
∫ dT 1ρ--- 1 T
ρ--- ρ∂
T∂------⎝ ⎠
⎛ ⎞p
+
p1
p2
∫ dp+=
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To successfully integrate Equation 10, the ANSYS CFX-Solver must be provided
thermodynamically consistent values of the equation of state, , and specific heat capacity,
. “Thermodynamically consistent” means that the coefficients of the differential terms of
Equation 9 must satisfy the exact differential property that:
or
The equation of state derivative within the second integral of Equation 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 to construct the property table,
and bases the number of points on an absolute error tolerance estimated using the
enthalpy and entropy derivatives.
Total Enthalpy Total enthalpy is expressed in terms of a static enthalpy and the flow kinetic energy:
(Eqn. 11)
ρ
cp
T 2
2
∂∂ h
⎝ ⎠⎜ ⎟⎛ ⎞
p p2
2
∂∂ h
⎝ ⎠⎜ ⎟⎛ ⎞
T=
T 2
2
∂∂ cp
⎝ ⎠⎜ ⎟⎛ ⎞
p
1ρ--- ρ∂
T∂------⎝ ⎠
⎛ ⎞p
1ρ---
2Tρ2------- ρ∂
T∂------⎝ ⎠
⎛ ⎞p
+⎝ ⎠⎛ ⎞ T
ρ2-----
T 2
2
∂∂ ρ
⎝ ⎠⎜ ⎟⎛ ⎞
p+=
T
(p2,T2)(p1,T1)
p
h
21
htot hstat12-- U U⋅( )+=
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where is the flow velocity. When you use the total energy model the ANSYS CFX-Solver
directly computes total enthalpy, and static enthalpy is derived from this expression. In
rotating frames of reference the total enthalpy includes the relative frame kinetic energy.
For details, see Rotating Frame Quantities (p. 17).
DomainTemperature
The domain temperature, , is the absolute temperature at which an isothermal
simulation is performed. For details, see Isothermal (p. 8 in "ANSYS CFX-Solver Modeling
Guide").
StaticTemperature
The static temperature, , is the thermodynamic temperature, and depends on the
internal energy of the fluid. In ANSYS CFX, depending on the heat 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 and the constitutive relationship
for the material under consideration. The constitutive relation simply 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 general differential relationship for
enthalpy:
(Eqn. 12)
which is derived from the full differential form for changes in static 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 specific heat as a function of
temperature is defined by:
(Eqn. 13)
When the solver calculates static enthalpy, either directly or from total enthalpy, you can
back static temperature out of this relationship. When varies with temperature, the
ANSYS CFX-Solver builds an enthalpy table and static temperature is backed out by
inverting the table.
Material with Variable Density and Specific HeatTo properly handle materials with an equation of state and specific heat that vary as
functions of temperature and pressure, the ANSYS CFX-Solver needs to know enthalpy as a
function of temperature and pressure, .
U
Tdom
T stat
ρ cp
hstat href– cp T stat T ref–( )=
T ref 0 K[ ]= href 0 J kg( )⁄[ ]=
hstat href– cp T( ) TdT ref
T stat
∫=
cp
h T p,( )
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can be provided as a table using, for example, an RGP file. If a table is not
pre-supplied, and the equation of state and specific heat are given by CEL expressions or
CEL user functions, the ANSYS CFX-Solver will calculate by integrating the full
differential definition of enthalpy change.
Given the knowledge of and that the ANSYS CFX-Solver calculates both static
enthalpy and static pressure from the flow solution, you can calculate static temperature by
inverting the enthalpy table:
(Eqn. 14)
In this case, you know , from solving the flow and you calculate by table
inversion.
TotalTemperature
The total temperature is derived from the concept of total enthalpy and is computed exactly
the same way as static temperature, except that total enthalpy is used in the property
relationships.
Material with Constant Density and Specific Heat
If and are constant, then the total temperature and static temperature are equal
because incompressible fluids undergo no temperature change due to addition of kinetic
energy. This can be illustrated by starting with the total enthalpy form of the constitutive
relation:
(Eqn. 15)
and substituting expressions for Static Enthalpy and Total Pressure 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
ρ cp
htot href– cp T tot T ref–( )=
htot hstat12-- U U⋅( )+=
ptot pstat12--ρ U U⋅( )+=
T tot T stat=
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Ideal Gas with constant cp
For this case, enthalpy is only a function of temperature and the constitutive relation is:
(Eqn. 19)
which, if one substitutes the relation between static and total enthalpy, yields:
(Eqn. 20)
If the turbulence model is employed, turbulence kinetic energy, , is added to the
total enthalpy, giving the modified total temperature for constant :
(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 static temperature
except that total enthalpy and total pressure are used as inputs into the enthalpy table:
(Eqn. 23)
In this case you know , and you want to calculate , but you do not know . So,
before calculating the total temperature, you need to compute total 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 of thermodynamics:
(Eqn. 24)
h href– cp T T ref–( )=
T tot T statU U⋅2cp
-------------+=
k ε– k
cp
T tot T statU U⋅2cp
------------- kcp-----+ +=
htot href– cp T( ) TdT ref
T tot
∫=
htot href– h T tot ptot,( ) h T ref pref,( )–=
htot T tot ptot
Tds dh dpρ------–=
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 constitutive relationship for the material, you
can arrive at various forms for calculating the changes in entropy 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 are given by:
(Eqn. 26)
and when this is substituted into the second law gives the following expression for changes
in entropy as a function of temperature only:
(Eqn. 27)
which when integrated gives an analytic formula for changes in entropy:
(Eqn. 28)
Ideal Gas with constant cp or cp = f(T)
For ideal gases changes in entropy are given by the following equation:
(Eqn. 29)
which for general functions for the solver computes an entropy table as a function of
both temperature and pressure. In the simplified case when is a constant, then an
analytic formula is used:
(Eqn. 30)
ds dhT------ dp
ρT-------–=
dh cpdT dpρ------+=
ds cpdTT
-------=
s sref– cpT
T ref---------⎝ ⎠
⎛ ⎞log=
s sref–cp T( )
T-------------- Td
T ref
T∫ R p
pref--------⎝ ⎠
⎛ ⎞log–=
cp
cp
s sref– cpT
T ref---------⎝ ⎠
⎛ ⎞log R ppref--------⎝ ⎠
⎛ ⎞log–=
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Material with Variable Density and Specific HeatThis is the most general case handled by the ANSYS CFX-Solver. The entropy function,
, is calculated by integrating the full differential form for entropy as a function of
temperature and pressure. Instead of repetitively performing this integration the ANSYS
CFX-Solver computes a table of values at a number of temperature and pressure points. The
points are chosen such that the approximation error for the entropy function is minimized.
The error is estimated using derivatives of entropy with respect to temperature and
pressure. Expressions for the derivatives are found by substituting the formula for general
enthalpy changes into the second law to get the following expression for changes in
entropy with temperature and pressure:
(Eqn. 31)
which when compared with the following differential form for changes in entropy:
(Eqn. 32)
gives that:
(Eqn. 33)
(Eqn. 34)
The derivative of entropy with respect to temperature is exactly evaluated, while the
derivative with respect to pressure must be computed by numerically differentiating the
equation of state. Note that when properties are specified as a function of temperature and
pressure using CEL expressions the differential terms in Equation 31 must also satisfy the
exact differential relationship:
See the previous section on Static Enthalpy for more details.
s T p,( )
dscpT-----dT 1
ρ2----- ρ∂
T∂------⎝ ⎠
⎛ ⎞pdp+=
ds s∂T∂
------⎝ ⎠⎛ ⎞
pdT s∂
p∂------⎝ ⎠
⎛ ⎞T
dp+=
s∂T∂
------⎝ ⎠⎛ ⎞
p
cpT-----=
s∂p∂
------⎝ ⎠⎛ ⎞
T
1
ρ2----- ρ∂
T∂------⎝ ⎠
⎛ ⎞p
=
T 2
2
∂∂ s
⎝ ⎠⎜ ⎟⎛ ⎞
p p2
2
∂∂ s
⎝ ⎠⎜ ⎟⎛ ⎞
T=
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Unless an externally provided table is supplied, an entropy table is built by using the and
functions supplied by the user and integrating the general differential form:
(Eqn. 35)
To calculate total pressure, you also need to evaluate entropy as a function of enthalpy and
pressure, rather than temperature and pressure. For details, see Total Pressure (p. 14).
The recipe to do this is essentially the same as for the temperature and pressure recipe. First,
you start with differential form for :
(Eqn. 36)
and comparing this with a slightly rearranged form of the second law:
(Eqn. 37)
you get that:
(Eqn. 38)
(Eqn. 39)
In this case, the entropy derivatives with respect to and can be evaluated from the
constitutive relationship (getting from and by table inversion) and the
equation of state . Points in the table are evaluated by performing the
following integration:
(Eqn. 40)
over a range of and , which are determined to minimize the integration error.
Total Pressure The total pressure, , is defined as the pressure that would exist at a point if the fluid was
brought instantaneously to rest such that the dynamic energy of the flow converted to
pressure without losses. The following three sections describe how total pressure is
ρ
cp
s2 s1–cpT-----dT
T1
T2
∫ 1
ρ2----- ρ∂
T∂------⎝ ⎠
⎛ ⎞
p1
p2
∫ pdp+=
s h p,( )
ds s∂h∂
------⎝ ⎠⎛ ⎞
pdh s∂
p∂------⎝ ⎠
⎛ ⎞hdp+=
ds dhT------ dp
ρT-------–=
s∂h∂
------⎝ ⎠⎛ ⎞
p
1T---=
s∂p∂
------⎝ ⎠⎛ ⎞
h
1ρT-------–=
h p
h T p,( ) T h p
r T p,( ) s h p,( )
s sref–1T--- hd
href
h
∫ 1ρT-------
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 of liquids and low speed gas flows, the total pressure
is given by Bernoulli’s equation:
(Eqn. 41)
which is the sum of the static and dynamic pressures.
Ideal GasesWhen the flow is compressible, total pressure is computed by starting with the second law
of thermodynamics assuming that fluid state variations are locally isentropic so that you get:
(Eqn. 42)
The left hand side of this expression is determined by the constitutive relation and the right
hand side by the equation of state. For an ideal gas, the constitutive relation and equation
of state are:
(Eqn. 43)
(Eqn. 44)
which when substituted into the second law and assuming no entropy variations gives:
(Eqn. 45)
where and are the static and total temperatures respectively (calculation of these
quantities was described in two previous sections, Static Temperature and Total
Temperature). If is a constant, then the integral can be exactly evaluated, but if varies
with temperature, then the integral is numerically evaluated using quadrature.
• For details, see Entropy (p. 11).
• For details, see Ideal Gas Equation of state (p. 25).
ptot pstat12--ρ U U⋅( )+=
dh dpρ------=
dh cp T( )dT=
ρ 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 pressure calculations in this case are somewhat more involved. You need to calculate
total pressure given the static enthalpy and static pressure, which the ANSYS CFX-Solver
solves for. You also want to assume that the local state variations, within a control volume
or on a boundary condition, are isentropic.
Given that you know (from integrating the differential form for enthalpy) 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 trivial procedure:
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 often simply 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,( )
ρ 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 i∂x j∂
---------U j∂xi∂
---------+⎝ ⎠⎛ ⎞=
sstrnr 2U i∂x 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 a function of this scalar shear
strain rate.
Rotating FrameQuantities
The velocity in the rotating frame of reference is defined as:
(Eqn. 49)
where is the angular velocity, is the local radius vector, and is velocity in the
stationary frame of reference.
Incompressible FluidsFor incompressible flows, the total pressure is defined as:
(Eqn. 50)
where is static pressure. The stationary frame total pressure is defined 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 Total Pressure. First you start
with the relative total enthalpy, rothalpy and stationary frame total enthalpy:
(Eqn. 52)
(Eqn. 53)
(Eqn. 54)
U x U y U z
sstrnr 2U x∂x∂
----------⎝ ⎠⎛ ⎞
2 U y∂y∂
----------⎝ ⎠⎛ ⎞
2 U z∂z∂
---------⎝ ⎠⎛ ⎞
2+ +
⎩ ⎭⎨ ⎬⎧ ⎫
=
U x∂y∂
----------U y∂x∂
----------+⎝ ⎠⎛ ⎞
2 U x∂z∂
----------U z∂x∂
---------+⎝ ⎠⎛ ⎞
2 U y∂z∂
----------U z∂y∂
---------+⎝ ⎠⎛ ⎞
2+ + +
12--
Urel Ustn ω R×–=
ω R Ustn
ptot pstat12--ρ Urel Urel⋅ ω R× ω R×⋅( )–( )+=
pstat
ptot,stn pstat12--ρ Ustn Ustn⋅( )+=
htot hstat12-- Urel Urel⋅( )+=
I hstat12-- Urel Urel⋅ ω R× ω 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 relative kinetic energy.
Important: Rothalpy is not a positive definite quantity. If the rotation velocity is large,then the last term can be significantly larger than the static enthalpy plus the rotating framekinetic energy. In this case, it is possible that total temperature and rotating total pressureare undefined and will be clipped at internal table limits or built in lower bounds. However,this is only a problem for high angular velocity rotating systems.
If you again assume an ideal gas equation of state with variable specific heat capacity you
can compute relative total temperature, total temperature and stationary frame total
temperature using:
(Eqn. 55)
and:
(Eqn. 56)
and:
(Eqn. 57)
where all the total temperature quantities are obtained by inverting the enthalpy table. If
is constant, then these values are obtained directly from these definitions:
(Eqn. 58)
(Eqn. 59)
(Eqn. 60)
At this point, given , , , and you can compute relative total
pressure, total pressure or stationary frame total pressure using the relationship given in the
section describing total pressure. For details, see Total Pressure (p. 14).
The names of the various total enthalpies, temperatures, and pressures when visualizing
results in ANSYS CFX-Post or for use in CEL expressions is as follows.
hstat
htot
I( )
htot href– cp T( ) TdT ref
T tot,rel
∫=
I href– cp T( ) TdT ref
T tot
∫=
htot,stn href– cp T( ) TdT ref
T tot,stn
∫=
cp
T tot,rel T statUrel Urel⋅
2cp-----------------------+=
T tot T statUrel Urel⋅ ω R× ω 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, and pressures
The Mach Number and stationary frame Mach numbers are defined as:
(Eqn. 61)
(Eqn. 62)
where is the local speed of sound.
Material with Variable Density and Specific HeatRotating and stationary frame total temperature and pressure are calculated the same way
as described in Total Temperature and Total Pressure. The only changes in the recipes are
that rotating frame total pressure and temperature require rothalpy, , as the starting point
and stationary frame total pressure and temperature require stationary frame total
enthalpy, .
Courant number The Courant number is of fundamental importance for 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 of this expression where the
velocity and length scale are based on the mass flow into the control volume and the
dimension of the control volume.
Variable Long Variable Name Short Variable NameTotal Enthalpy htot
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
htot
Ihtot,stn
T tot,rel
T tot
T tot,stn
Ptot,rel
Ptot
Ptot,stn
MUrel
c------------=
MstnUstn
c-------------=
c
I
htot,stn
Courant u∆t∆x---------=
u t∆ x∆
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For explicit CFD methods, the timestep must be chosen such that the 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 the Courant 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 transient details.
ANSYS CFX uses the Courant number in a number of ways:
1. The timestep may be chosen adaptively based on a Courant number condition (e.g., toreach RMS or Courant number of 5).
2. For transient runs, the maximum and RMS Courant numbers are written 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 used throughout the ANSYS CFX-Solver
documentation.
The vectoroperators
Assume a Cartesian coordinate system in which , and are unit vectors in the three
coordinate directions. is defined such that:
(Eqn. 64)
Gradient operator
For a general scalar function , the gradient of is defined by:
(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∂
∂, ,=
φ x y z, ,( ) φ
φ∇ φ∂x∂
------i φ∂y∂
------ j φ∂z∂
------k+ +=
U x y z, ,( )
UUx
U y
Uz
=
U
U∇•U x∂x∂
----------U y∂y∂
----------U z∂z∂
---------+ +=
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Dyadic operator
The dyadic operator (or tensor product) of two vectors, and , is defined as:
(Eqn. 68)
By using specific tensor notation, the equations relating to each dimension can be
combined into a single equation. Thus, in the specific tensor notation:
(Eqn. 69)
Matrixtransposition
The transpose of a matrix is defined by the operator . For example, 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 in this documentation, the following may
help you if you are used to index notation.
U V
U V⊗U xV x U xV y U xV z
U yV x U yV y U yV z
U zV x U zV y U zV z
=
ρU U⊗( )∇•
x∂∂ ρUxUx( )
y∂∂ ρU yUx( )
z∂∂ ρUzUx( )+ +
x∂∂ ρUxU y( )
y∂∂ ρU yU y( )
z∂∂ ρUzU y( )+ +
x∂∂ ρUxUz( )
y∂∂ ρU yUz( )
z∂∂ ρUzUz( )+ +
=
T
φ∇
φ∂x∂
------
φ∂y∂
------
φ∂z∂
------
=
φ∇T φ∂
x∂------ φ∂
y∂------ φ∂
z∂------=
δ1 0 00 1 00 0 1
=
Basic Solver Capability Theory: Governing Equations
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In index notation, the divergence operator can be written:
(Eqn. 73)
where the summation convention is followed, i.e., the index is summed over the three
components.
The quantity can be represented by (when and are vectors), or by
(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 the divergence operator are
derivatives with respect to the same coordinate as the first listed 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 unsteady Navier-Stokes equations in their
conservation form.
If you are new to CFD, review the introduction. For details, see Computational Fluid
Dynamics (p. 1 in "ANSYS CFX Introduction").
A list of recommended books on CFD and related subjects is available. For details, see
Further Background Reading (p. 6 in "ANSYS CFX Introduction").
For all the following equations, static (thermodynamic) quantities are given unless
otherwise stated.
U∇•U i∂xi∂
---------=
i
U V⊗ U iV j U V
U iV jk U V
ρU U⊗( )∇•
xi∂∂ ρU iU j( )
ρU U⊗( )∇•
xi∂∂ ρU iU j( )
x j∂∂ ρU iU j( )
a b• a b aijbij
Basic Solver Capability Theory: Governing Equations
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Transport Equations
In this section, the instantaneous equation of mass, momentum, and energy conservation
are presented. For turbulent flows, the instantaneous equations are 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 energy conservation 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 enthalpy by:
(Eqn. 81)
The term represents the work due to viscous stresses and is called the viscous
work term.
The term represents the work due to external momentum sources and is currently
neglected.
The ThermalEnergy Equation
An alternative form of the energy equation, which is suitable for low-speed flows, is also
available. To derive it, an equation is required for the mechanical energy .
(Eqn. 82)
The mechanical energy equation is derived by taking the dot product of with the
momentum equation (Eqn. 78):
(Eqn. 83)
ρ∂t∂
------ ρU( )∇•+ 0=
ρU( )∂t∂
---------------- ρU U⊗( ) ∇p ∇ τ•+– SM+=∇•+
τ
τ µ ∇U ∇U( )T 23--δ∇ U•–+⎝ ⎠
⎛ ⎞=
ρhtot( )∂t∂
------------------- p∂t∂
------ ρUhtot( )∇•+– λ∇T( ) ∇ U τ•( )• U SM S+•+ E+∇•=
htot h T p,( )
htot h 12--U2
+=
∇ U τ•( )•
U SM•
K
K 12--U2
=
U
ρK( )∂t∂
--------------- ρUK( ) U– ∇p• U ∇ τ•( )• U SM•+ +=∇•+
Basic Solver Capability Theory: Governing Equations
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Subtracting this equation from the total energy equation (Eqn. 80) yields the thermal energy
equation:
(Eqn. 84)
The term is always negative and is called the viscous dissipation. Finally, the static
enthalpy is related to the internal energy by:
(Eqn. 85)
So (Eqn. 84) can be simplified to:
(Eqn. 86)
The term is currently neglected, although it may be non-zero for variable-density
flows. This is the thermal energy equation solved by ANSYS CFX.
Please note the following guidelines regarding use of the thermal energy equation:
• Although the thermal energy equation solves for , this variable is still called staticenthalpy in ANSYS CFX-Post.
• The thermal energy equation is meant to be used for flows which are low speed andclose to constant density
• The thermal energy equation is particularly suited for liquids, since compressibilityeffects are minor. In addition, the total energy equation may experience robustness
problems due to the pressure transient and the contribution to enthalpy.
• For materials which have variable specific heats (e.g., set as a CEL expression or using an
RGP table or Redlich Kwong equation of state) the solver includes the contribution
in the enthalpy tables. This is inconsistent, because the variable is actually internal
energy. For this reason, the thermal energy equation should not be used in this
situation, particularly for subcooled liquids.
Equations of State
In ANSYS CFX, the flow solver calculates pressure and static enthalpy. Finding density
requires that you select the thermal equation of state and finding temperature requires that
you select the constitutive relation. The selection of these two relationships is not
necessarily independent and is also a modeling choice.
The thermal equation of state is described as a function of both temperature and pressure:
(Eqn. 87)
ρh( )∂t∂
-------------- p∂t∂
------ ρUh( ) λ∇T( ) U ∇p τ:∇U+• SE++∇•=∇•+–
τ:∇U
h e
h e Pρ---+=
ρe( )∂t∂
------------- ρUe( ) λT( ) p∇ U τ:∇U+• SE++∇•=∇•+
p∇ U•
e
P ρ⁄
P ρ⁄
ρ ρ p T,( )=
Basic Solver Capability Theory: Governing Equations
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The specific heat capacity, , may also be described as a function of temperature and
pressure:
(Eqn. 88)
For an Ideal Gas, the density is defined by the Ideal Gas Law and, in this case, can be a
function of only temperature:
(Eqn. 89)
Important: When or are also functions of an algebraic additional variable, in addition
to temperature and pressure, then changes of that additional variable are neglected in theenthalpy and entropy functions. However, if that additional variable is itself only dependenton pressure and temperature, then the effects will be correctly accounted for.
Ideal GasEquation ofstate
For an Ideal Gas, the relationship is described by the Ideal Gas Law:
(Eqn. 90)
where is the molecular weight of the gas, and is the universal gas constant.
Real Gas andLiquidEquations ofState
In the current version of ANSYS CFX the Redlich Kwong equation of state is available as a
built-in option for simulating real gases. It is also available through several pre-supplied
CFX-TASCflow RGP files. The Vukalovich Virial equation of state is also available but currently
only by using CFX-TASCflow RGP tables.
Redlich Kwong Gas PropertiesThe Redlich-Kwong equation of state was first published in 1949 and is considered one of
the most accurate two parameter corresponding states equations of state. This equation of
state is quite useful from an engineering standpoint because it only requires that the user
know the fluid critical temperature and pressure. More recently, Aungier (1995) [96] has
modified the Redlich-Kwong equation of state so that it provides much better accuracy near
the critical point. The Aungier form of this equation of state is used by ANSYS CFX and is
given by:
(Eqn. 91)
cp
cp cp p T,( )=
cp
cp cp T( )=
ρ cp
ρw p pref+( )
R0T---------------------------=
w R0
p RTν b– c+-------------------- a T( )
ν ν b+( )--------------------–=
Basic Solver Capability Theory: Governing Equations
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This form differs from the original by the additional parameter , which is added to improve
the behavior of isotherms near the critical point, as well as the form of . The
parameters , , and in (Eqn. 91) are given by:
(Eqn. 92)
(Eqn. 93)
(Eqn. 94)
(Eqn. 95)
In the expression for , the standard Redlich Kwong exponent of has been
replaced by a general exponent . Optimum values of depend on the pure substance.
Aungier (1995) [96] presented values for twelve experimental data sets to which he
provided a best fit polynomial for the temperature exponent in terms of the acentric
factor, :
(Eqn. 96)
The acentric factor must be supplied when running the Redlich Kwong model and is
tabulated for many common fluids in Poling et al [84]. If you do not know the acentric factor,
or it is not printed in a common reference, it can be estimated using knowledge of the
critical point and the vapor pressure curve with this formula:
(Eqn. 97)
where the vapor pressure, , is calculated at . In addition to the critical point
pressure, this formula requires knowledge of the vapor pressure as a function of
temperature.
In order to provide a full description of the gas properties, the flow solver must also calculate
enthalpy and entropy. These are evaluated using slight variations on the general
relationships for enthalpy and entropy which were presented in the previous section on
variable definitions. The variations depend on the zero pressure, ideal gas, specific heat
c
a T( )
a b c
a a0TTc------⎝ ⎠
⎛ ⎞ n–=
a00.42747R2Tc
2
pc---------------------------------=
b0.08664RTc
pc-----------------------------=
cRTc
pca0
νc νc b+( )-------------------------+
------------------------------------- b νc–+=
a T( ) n 0.5=
n n
n
ω
n 0.4986 1.2735ω 0.4754ω2+ +=
ω log– 10pvpc-----⎝ ⎠
⎛ ⎞ 1–=
pv T 0.7Tc=
Basic Solver Capability Theory: Governing Equations
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capacity and derivatives of the equation of state. The zero pressure specific heat capacity
must be supplied to ANSYS CFX while the derivatives are analytically evaluated from (Eqn.
91).
Internal energy is calculated as a function of temperature and volume ( , ) by integrating
from the reference state ( , ) along path 'amnc' (see diagram below) to the required
state ( , ) using the following differential relationship:
(Eqn. 98)
First the energy change is calculated at constant temperature from the reference volume to
infinite volume (ideal gas state), then the energy change is evaluated at constant volume
using the ideal gas . The final integration, also at constant temperature, subtracts the
energy change from infinite volume to the required volume. In integral form, the energy
change along this path is:
(Eqn. 99)
Once the internal energy is known, then enthalpy is evaluated from internal energy:
T v
T ref vref
T v
ud cv Td T pdTd
------⎝ ⎠⎛ ⎞
vp–⎝ ⎠
⎛ ⎞ vd+=
cv
u T v,( ) u T ref vref,( )– =
T pdTd
------⎝ ⎠⎛ ⎞
vp–⎝ ⎠
⎛ ⎞ vT refd
vref
∞
∫ cv0 TdT ref
T
∫ T pdTd
------⎝ ⎠⎛ ⎞
vp–⎝ ⎠
⎛ ⎞ vTdv
∞
∫–+
h u pν+=
Basic Solver Capability Theory: Governing Equations
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The entropy change is similarly evaluated:
(Eqn. 100)
where is the zero pressure ideal gas specific heat capacity. By default ANSYS CFX uses a
4th order polynomial for this and requires that coefficients of that polynomial are available.
These coefficients are tabulated in various references including Poling et al [84].
In addition, a suitable reference state must be selected to carry out the integrations. The
selection of this state is arbitrary, and can be set by the user, but by default, ANSYS CFX uses
the normal boiling temperature (which is provided) as the reference temperature and the
reference pressure is set to the value of the vapor pressure evaluated using Equation 106 at
the normal boiling point. The reference enthalpy and entropy are set to zero at this point by
default, but can also be overridden if desired.
Other properties, such as the specific heat capacity at constant volume, can be evaluated
from the internal energy as:
(Eqn. 101)
where is the ideal gas portion of the internal energy:
(Eqn. 102)
specific heat capacity at constant pressure, , is calculated from using:
(Eqn. 103)
s T v,( ) s T ref vref,( )– =
pdTd
------⎝ ⎠⎛ ⎞
vvT ref
dvref
∞
∫cv0T------ Td
T ref
T
∫ R ppref--------⎝ ⎠
⎛ ⎞ln–pdTd
------⎝ ⎠⎛ ⎞
vvTd
v
∞
∫–+
cpo
cνu∂T∂
------⎝ ⎠⎛ ⎞
ν
u0∂T∂
--------n n 1+( )a
bT------------------------ log 1 b
v--+⎝ ⎠
⎛ ⎞–= =
u0
u0 u uref– cpo T( ) R–( )T ref
T
∫ dT= =
cp cv
cp cv νT β2
κ-----+=
Basic Solver Capability Theory: Governing Equations
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where and are the volume expansivity and isothermal compressibility, respectively.
These two values are functions of derivatives of the equation of state and are given by:
(Eqn. 104)
(Eqn. 105)
Redlich Kwong Saturated Vapor PropertiesWhen running calculations with liquid condensing out of the vapor (equilibrium phase
change or Eulerian thermal phase change model) or dry calculations, the flow solver needs
to know the form of the vapor pressure curve. Pressure and temperature are not
independent in the saturation dome so vapor saturation properties are evaluated by first
assuming an equation which gives the dependence of vapor pressure on temperature, and
then substituting that into the equation of state.
For materials that use the Redlich Kwong equation of state ANSYS CFX approximates the
vapor pressure curve using a form given by Aungier (2000) [65]:
(Eqn. 106)
Vapor saturation properties are calculated by evaluating the equation of state and
constitutive relations along the saturation curve.
Redlich Kwong Liquid PropertiesIt is possible to derive liquid densities directly from the Redlich Kwong equation, however,
this is not desirable because the Redlich Kwong equation is very inaccurate in the
compressed liquid regime.
Instead, when you select to use the Redlich Kwong equation of state for a liquid, the
properties are assumed to vary along the vapor pressure curve as a function of saturation
temperature. These properties are approximate and should only be used when the amount
of liquid in your calculation will be small. For example, they work well with the equilibrium
condensation model or non-equilibrium small droplet phase change model.
To derive the liquid enthalpy and entropy, such that they are completely consistent with the
gas phase, requires all the same data as is provided for the gas phase: the critical point data,
the acentric factor and the zero pressure specific heat coefficients.
β κ
β
p∂T∂
------⎝ ⎠⎛ ⎞
ν
ν p∂ν∂
------⎝ ⎠⎛ ⎞
T
------------------–=
κ 1
ν p∂ν∂
------⎝ ⎠⎛ ⎞
T
------------------–=
log10pvpc-----⎝ ⎠
⎛ ⎞ 73-- 1 ω+( ) 1
TcT------–⎝ ⎠
⎛ ⎞=
Basic Solver Capability Theory: Governing Equations
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To calculate saturated liquid densities, an alternative equation of state originally published
by Yamada and Gunn (1973), is used by ANSYS CFX that gives liquid specific volume as a
function of temperature:
(Eqn. 107)
This equation is convenient because it only requires knowledge of the critical volume and
temperature as well as the acentric factor. The valid temperature range for the liquid
equation of state is 0.4 < < 0.99 . The solver will clip the temperature used in this
equation to that range.
Saturated liquid enthalpy is calculated using knowledge of the gas saturation enthalpy and
the following equation:
(Eqn. 108)
where the enthalpy of vaporization, , is given by the following expression taken from
Poling et al [84]:
(Eqn. 109)
Saturated liquid entropy can easily be derived using the second law and the gas saturation
entropy:
(Eqn. 110)
Prediction of liquid specific heat capacity with the Redlich Kwong equation has a similar
problem to the liquid density, so ANSYS CFX uses an alternative form presented by Aungier
(2000):
(Eqn. 111)
which requires knowledge of the zero pressure heat capacity coefficients, as well as the
acentric factor. For the saturated liquid it is assumed that .
Vukalovich VirialThe virial equation of state used in ANSYS CFX was developed by Vukalovich and is given by:
(Eqn. 112)
ν νc 0.29056 0.08775ω–( )1 T T c⁄–( )2 7⁄
=
Tc T Tc
hf,s hg,s hfg–=
hfg
hfgRTc---------- 7.08 1 T
Tc------–⎝ ⎠
⎛ ⎞ 0.35410.95ω 1 T
Tc------–⎝ ⎠
⎛ ⎞ 0.456+=
sf sghfgT------–=
cp cp0 0.5 2.2ω+( ) 3.67 11.64 1 TTc------–⎝ ⎠
⎛ ⎞ 40.634 1 T
Tc------–⎝ ⎠
⎛ ⎞ 1–+ +⎝ ⎠
⎛ ⎞+=
cp cv=
p ρRT B1 B2ρ B3ρ2 B4ρ3+ + +( )=
Basic Solver Capability Theory: Governing Equations
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This equation of state is known to be valid over pressure and temperature ranges of 0.01 bar
to 100 bar and 273.15 K to 1000 K.
IAPWS Equation of StateThe IAPWS-IF97 database represents an accurate equation of state for water and steam
properties. The database is fully described elsewhere [125], but a summary will be provided
in this section. The IAPWS database uses formulations for five distinct thermodynamic
regions for water and steam, namely:
• subcooled water (1)
• supercritical water/steam (2)
• superheated steam (3)
• saturation data (4)
• high temperature steam (5)
Figure 1 Regions and Equations of IAPWS-IF97
Region 5 has not been implemented in ANSYS CFX because it represents a thermodynamic
space at very high temperatures (1073.15 - 2273.15 K) and reasonably low pressures (0-10
MPa) that can be adequately described using other property databases already in ANSYS
CFX (i.e., Ideal Gas EOS with NASA specific heat and enthalpy). Furthermore, since this region
is not defined for pressures up to 100 MPa, as is the case for regions 1, 2 and 3, problems
arise in filling out the pressure-temperature space in the tables when temperatures exceed
1073.15 K and pressures exceed 10 MPa. The database implemented in ANSYS CFX therefore
covers temperatures ranging from 273.15 to 1073.15 K and pressures ranging from 611 Pa
to 100 MPa.
The reference state for the IAPWS library is the triple point of water. Internal energy, entropy
and enthalpy are all set to zero at this point.
Tref = 273.16 K, Pref = 611.657 Pa, uliquid = 0 J/kg, sliquid = 0 J/kg/K, hliquid = 0 J/kg
Basic Solver Capability Theory: Governing Equations
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In ANSYS CFX, the analytical equation of state is used to transfer properties into tabular
form, which can be evaluated efficiently in a CFD calculation. These IAPWS tables are
defined in terms of pressure and temperature, which are then inverted to evaluate states in
terms of other property combinations (such as pressure/enthalpy or entropy/enthalpy).
When developing the IAPWS database for ANSYS CFX, therefore, properties must be
evaluated as functions of pressure and temperature. For the most part, this involves a
straightforward implementation of the equations described in the IAPWS theory [125].
Region 4 involves saturation data which only uses pressure or temperature information.
However, some difficulties are encountered when evaluating the properties around Region
3 (near the critical point), where the EOS is defined explicitly in terms of density and
temperature. In this region, the density must be evaluated using Newton-Raphson iteration.
This algorithm is further complicated in that the EOS is applicable on both the subcooled
liquid and superheated vapor side leading up to critical conditions. Therefore, depending
on the pressure-temperature state, one may be evaluating a subcooled liquid or a
superheated vapor with the same EOS. To apply the Newton-Raphson scheme in a reliable
way, one must detect on which side of the saturation dome the pressure-temperature state
applies, and apply an appropriate initial guess. Such an iteration scheme, including logic for
an initial guess, has been implemented in CFX so that table generation around the critical
region is possible.
Metastable Superheated Liquid/Supercooled Vapor StatesThe IAPWS library also extends to metastable states, so that this equation of state is available
for nonequilibrium phase change models such as the Droplet Condensation Model. The EOS
for regions 1 and 3 in Figure 1 are stated to have reasonable accuracy for metastable states
close to the saturation line d liquid states [125]. However, the term “reasonable” is not
quantified, and therefore the degree to which the extrapolation of the EOS can be applied
is unknown.
In region 2, an additional set of equations have been developed for supercooled vapor
conditions under 10 MPa. These equations have been tuned to match the saturation data.
Above 10 MPa, the EOS for the superheated region can safely be extrapolated into
supercooled conditions, but it does not match smoothly with the specialized supercooled
equations below 10 MPa.
Numerical Testing to Delineate Metastable RegionsIn order to make the IAPWS database as robust as possible, numerical testing has been done
to determine approximate metastable vapor/liquid spinodal lines. Figure 2 is given to
demonstrate these spinodal lines which are essentially boundaries up to which metastable
conditions can exist. These would be defined similar to saturation curves as functions of
Basic Solver Capability Theory: Buoyancy
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either temperature or pressure. The IAPWS database tables are always generated up to
these limits regardless of what flow models are specified (i.e., equilibrium or
non-equilibrium) and thus allow nonequilibrium phase change models to be applied.
Figure 2 Spinodal Limits Built Into Tables
Conjugate Heat Transfer
ANSYS CFX allows you to create solid regions in which the equations for heat transfer are
solved, but with no flow. This is known as conjugate heat transfer, and the solid regions are
known as solid domains.
Within solid domains, the conservation of energy equation is simplified since there is no
flow inside a solid, thus conduction is the only mode of heat transfer. The heat conduction
through the solid has the following transport equation:
(Eqn. 113)
where , and are the density, specific heat capacity and thermal conductivity of the
solid, respectively.
Additional information on plotting variables at a solid-fluid interface is available. For details,
see Solid-Fluid Interface Variable Values (p. 33 in "ANSYS CFX-Post User's Guide").
Buoyancy
For buoyancy calculations, a source term is added to the momentum equations as follows:
(Eqn. 114)
t∂∂ ρcpT( ) ∇ λ∇T( ) SE+•=
ρ cp λ
SM,buoy ρ ρref–( )g=
Basic Solver Capability Theory: Buoyancy
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The density difference is evaluated using either the Full Buoyancy model or the
Boussinesq model, depending on the physics.
When buoyancy is activated, the pressure in the momentum equation excludes the
hydrostatic gradient due to . This pressure is related to the absolute pressure as follows:
(Eqn. 115)
where is a reference location. The reference location option is set under the Ref
Location option in ANSYS CFX-Pre and can be set to a particular location (Cartesian
Coordinates). For details, see Buoyancy: Option (p. 148 in "ANSYS CFX-Pre User's Guide"). If
unset (option set to Automatic), the solver defaults it to the centroid of a pressure-specified
boundary (if one exists), or to the pressure reference location (if no pressure-specified
boundary exists).
Absolute pressure is used to evaluate fluid properties which are functions of pressure, and
can also be visualized in ANSYS CFX-Post.
Full Buoyancy Model
For buoyancy calculations involving variable density, is evaluated directly. This
option is set automatically when the simulation involves multicomponent flow, multiphase
flow, or a fluid having density set as a function of pressure, temperature, or other field
variables.
Boussinesq Model
For buoyant flows where the density variation is driven only by small temperature
variations, the Boussinesq model is used. In this model, a constant reference density is
used for all terms other than the buoyancy source term. The buoyancy source term is
approximated as:
(Eqn. 116)
where is the thermal expansivity:
(Eqn. 117)
and is the buoyancy reference temperature.
ρ ρref–
ρref
pabs p pref ρref g r rref–( )+ +=
rref
ρ ρref–
ρref
ρ ρref– ρrefβ T T ref–( )–=
β
β 1ρ--- ∂ρ
∂T-------
p–=
T ref
Basic Solver Capability Theory: Multicomponent Flow
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Multicomponent Flow
The following topics will be discussed:
• Multicomponent Notation (p. 35)
• Scalar Transport Equation (p. 35)
• Algebraic Equation for Components (p. 37)
• Constraint Equation for Components (p. 37)
• Multicomponent Fluid Properties (p. 38)
• Energy Equation (p. 39)
• Multicomponent Energy Diffusion (p. 40)
Multicomponent Notation
Components are denoted using capital letters , , , etc. In general, a quantity
subscribed with , , , etc., refers to the value of the quantity corresponding to , ,
, etc. For example, the density ( ) of component would be written .
Scalar Transport Equation
For a multicomponent fluid, scalar transport equations are solved for velocity, pressure,
temperature and other quantities of the fluid. For details, see Governing Equations (p. 22).
However, additional equations must be solved to determine how the components of the
fluid are transported within the fluid.
The bulk motion of the fluid is modeled using single velocity, pressure, temperature and
turbulence fields. The influence of the multiple components is felt only through property
variation by virtue of differing properties for the various components. Each component has
its' own equation for conservation of mass. After Reynolds-averaging (see Turbulence
Models (π. 69 ιν ∀ΑΝΣΨΣ ΧΦΞ−Σολϖερ Τηεορψ Γυιδε∀)) this equation can be
expressed in tensor notation as:
(Eqn. 118)
where:
is the mass-average density of fluid component in the mixture, i.e., the mass of the
component per unit volume,
is the mass-average velocity field,
is the mass-average velocity of fluid component ,
A B C
A B C A B
C ρ B ρB
∂ρi∂t--------
∂ ρiU j( )∂x j
-------------------+
∂∂x j-------- ρi U ij U j–( ) ρi'' U j ''–( )– Si+=
ρi i
U j ρiU ij( ) ρ⁄∑=
U ij i
Basic Solver Capability Theory: Multicomponent Flow
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is the relative mass flux,
is the source term for component which includes the effects of chemical reactions.
Note that if all the equations (Eqn. 118) are summed over all components, the result is the
standard continuity equation,
(Eqn. 119)
since the reaction rates must sum to zero.
The relative mass flux term accounts for differential motion of the individual components.
This term may be modeled in a number of ways to include effects of concentration
gradients, a pressure gradient, external forces or a temperature gradient. Of these possible
sources of relative motion among the mixture components, the primary effect is that of
concentration gradient. The model for this effect gives rise to a diffusion-like term in (Eqn.
118).
(Eqn. 120)
The molecular diffusion coefficient, , is assumed to be equal to , where is the
Kinematic Diffusivity set on the Fluid Models tab for a domain in ANSYS CFX-Pre. For
details, see Fluid Models Tab (p. 149 in "ANSYS CFX-Pre User's Guide"). A detailed
description of the effects of the relative mass flux term and various models for it may be
found in reference [29].
Now, define the mass fraction of component to be:
(Eqn. 121)
Note that, by definition, the sum of component mass fractions over all components is 1.
Substituting (Eqn. 121) and (Eqn. 120) into (Eqn. 118), you have:
(Eqn. 122)
ρi U ij U j–( )
Si i
∂ρ∂t------
∂ ρU j( )∂x j
------------------+ 0=
Si
ρi U ij U j–( )Γiρ-----
∂ρi∂x j--------–=
Γi ρDi Di
i
Y iρiρ----=
∂ ρY i( )∂t
-----------------∂ ρU jY i( )
∂x j------------------------+
∂∂x j-------- Γi
∂Y i∂x j--------⎝ ⎠
⎛ ⎞ ∂∂x j-------- ρY i
′′U j′′( )– Si+=
Basic Solver Capability Theory: Multicomponent Flow
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The turbulent scalar fluxes are modeled using the eddy dissipation assumption as:
(Eqn. 123)
where is the turbulent Schmidt number. Substituting (Eqn. 123) into (Eqn. 122) and
assuming now that you have mass weighted averages of :
(Eqn. 124)
where:
(Eqn. 125)
(Eqn. 124) is simply a general advection-diffusion equation of the form common to the
equations solved for each of the other dependent variables in the fluid flow calculation.
Thus, it is convenient to solve for the in order to establish the composition of the fluid
mixture.
Algebraic Equation for Components
The specified equation is used to calculate the component mass fraction throughout the
solution domain.
Constraint Equation for Components
The ANSYS CFX-Solver solves mass fraction equations (either transport equations or
algebraic equations) for all but one of the components. The remaining component is known
as the constraint component because its mass fraction is determined by the constraint
equation:
(Eqn. 126)
The performance of the ANSYS CFX-Solver will not be affected by your choice of constraint
component.
ρY i′′U j
′′ µtSct------
∂Y i∂x j--------=
Sct
Y i
∂ ρY i( )∂t
-----------------∂ ρU jY i( )
∂x j------------------------+
∂∂x j-------- Γieff
∂Y i∂x j--------⎝ ⎠
⎛ ⎞ Si+=
ΓieffΓi
µtSct-------+=
Y i
NC 1–
Y ii A B C …, , ,=
N C
∑ 1=
Basic Solver Capability Theory: Multicomponent Flow
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Multicomponent Fluid Properties
The physical properties of general multicomponent mixtures are difficult to specify. The
default treatment in ANSYS CFX, Release 11.0 makes the assumption that the components
form an ideal mixture. For details, see Ideal Mixture (p. 15 in "ANSYS CFX-Solver Modeling
Guide").
Now consider a given volume of the fluid mixture. Let be the mass of component
present in this volume, then . The partial volume of component is defined to
be the volume, , that would be occupied by the given mass of the component at the
same (local) temperature and pressure as the mixture. The “thermodynamic density” of the
component, which results from evaluating its equation of state at the mixture temperature
and pressure, may be expressed as . Since the partial volumes of all
components must sum to the total volume, , you have:
(Eqn. 127)
or:
(Eqn. 128)
Thus, the mixture density may be calculated from the mass fractions and the
thermodynamic density of each component, which may require knowledge of the mixture
temperature and pressure, as well as an appropriate equation of state for each component.
Note carefully the distinction between and . The component mass density, , is a
quantity relating to the composition of the mixture, while the thermodynamic density,
, is a material property of the component.
An arbitrary constitutive fluid property may be calculated from:
(Eqn. 129)
V Mi i
ρi Mi V⁄= i
V i
ρi⟨ ⟩ Mi V i⁄=
V
1V iV-----
i A B…,=
N C
∑Mi ρi⟨ ⟩⁄
Mi ρi⁄---------------------
i A B…,=
N C
∑= =
ρiρi⟨ ⟩
----------i A B…,=
N C
∑ρY iρi⟨ ⟩
----------i A B…,=
N C
∑= =
1ρ---
Y iρi⟨ ⟩
----------i A B…,=
N C
∑=
Y i
ρi ρi⟨ ⟩ ρi
ρi⟨ ⟩
α Y i αii A B…,=
N C
∑=
Basic Solver Capability Theory: Multicomponent Flow
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where is the property value for fluid component . While it may appear anomalous at
first sight that density does not conform to this expression, the specific volume (volume per
unit mass, i.e., ) does indeed conform, as can be seen by considering (Eqn. 129).
Properties that may be evaluated for a multicomponent mixture using (Eqn. 129) include
the laminar viscosity , the specific heat at constant volume , the specific heat at
constant pressure , and the laminar thermal conductivity .
Energy Equation
Recall that (Eqn. 5) is the Reynolds-averaged conservation equation for energy of a single
component fluid. Extending this equation for multicomponent fluids involves adding an
additional diffusion term to the energy equation:
(Eqn. 130)
For turbulent flow, this term is Reynolds-averaged (see Turbulence Models (π. 69 ιν∀ΑΝΣΨΣ ΧΦΞ−Σολϖερ Τηεορψ Γυιδε∀)) giving:
(Eqn. 131)
This expression introduces several terms involving the fluctuations of diffusion coefficient,
component enthalpy and species concentration. Under certain circumstances, the
fluctuating components could be an important component of the diffusion process.
However, adequate models are not available within the existing turbulence model to
account for these effects. Thus, only the mean component is retained in the current version
of ANSYS CFX.
The implemented conservation of energy equation for multicomponent fluids involves only
mean scalar components and is expressed as:
(Eqn. 132)
αi i
1 ρi⟨ ⟩⁄
µ cv
cp λ
∂∂x j-------- Γihi
∂Y i∂x j--------
i
N C
∑
∂∂x j-------- Γi Γi
′′+( ) hi hi
′′+( )
∂ Y i Y i′′
+( )∂x j
--------------------------i
N C
∑
∂∂t----- ρH( ) ∂P
∂t------–
∂∂x j-------- ρU jH( )+
∂∂x j-------- λ ∂T
∂x j-------- Γihi
∂Y i∂x j--------
i
N C
∑µtPrt------ ∂h
∂x j--------+ +
⎝ ⎠⎜ ⎟⎛ ⎞
SE+=
Basic Solver Capability Theory: Additional Variables
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Multicomponent Energy Diffusion
The energy equation can be simplified in the special case that all species diffusivities are the
same and equal to thermal conductivity divided by specific heat capacity,
(Eqn. 133)
This equation (Eqn. 133) holds when the Lewis number is unity for all components:
. For turbulent flow, assuming for all components is usually
just as good as the common practice of using the fluid viscosity for the default component
diffusivity (unity Schmidt number, ). For , the energy equation
(Eqn. 132) simplifies exactly to the following:
(Eqn. 134)
This equation (Eqn. 134) has the advantage that only a single diffusion term needs to be
assembled, rather than one for each component plus one for heat conduction. This can
significantly reduce numerical cost, in particular when the fluid consists of a large number
of components.
Additional Variables
There are several types of additional variable equations supported by ANSYS CFX-Solver:
• Transport Equation (p. 40)
• Diffusive Transport Equation (p. 41)
• Poisson Equation (p. 41)
• Algebraic Equation (p. 42)
Transport Equation
The general form of the transport equation for an additional variable is:
(Eqn. 135)
where:
• is the mixture density, mass per unit volume
• is the conserved quantity per unit volume, or concentration
• is the conserved quantity per unit mass
Γi Γ λcp-----= =
Lei λ cpΓi( )⁄ 1= = Lei 1=
Sci µ Γi⁄ 1= = Lei 1=
t∂∂ ρH( ) P∂
t∂------–
x j∂∂ ρU jH( )+
x j∂∂ λ
cp-----
µtPrt-------+⎝ ⎠
⎛ ⎞ h∂x j∂
------- SE+=
ρφ( )∂t∂
-------------- ρUφ( )∇•+ ρDΦ∇φ( )∇• Sφ+=
ρ
Φ
φ Φ ρ⁄=
Basic Solver Capability Theory: Additional Variables
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• is a volumetric source term, with units of conserved quantity per unit volume per
unit time
• is the kinematic diffusivity for the scalar
For turbulent flow, this equation is Reynolds-averaged (see Τυρβυλενχε Μοδελσ (π. 69 ιν∀ΑΝΣΨΣ ΧΦΞ−Σολϖερ Τηεορψ Γυιδε∀) and becomes:
(Eqn. 136)
where:
• is the turbulence Schmidt number
• is the turbulence viscosity
Diffusive Transport Equation
The general form of the diffusive transport equation for an additional variable (non-reacting
scalar) is:
(Eqn. 137)
where:
• is the mixture density, mass per unit volume.
• is the conserved quantity per unit volume, or concentration.
• is the conserved quantity per unit mass.
• is a volumetric source term, with units of conserved quantity per unit volume per
unit time
• is the kinematic diffusivity for the scalar.
Poisson Equation
The general form of the Poisson equation for an additional variable (non-reacting scalar) is:
(Eqn. 138)
where:
• is the mixture density, mass per unit volume.
• is the conserved quantity per unit volume, or concentration.
• is the conserved quantity per unit mass.
Sφ
DΦ
ρφ( )∂t∂
-------------- ρUφ( )∇•+ ρDΦµtSct-------+⎝ ⎠
⎛ ⎞ ∇φ⎝ ⎠⎛ ⎞∇• Sφ+=
Sct
µt
ρφ( )∂t∂
-------------- ρDΦ∇φ( )∇• Sφ+=
ρ
Φ
φ Φ ρ⁄=
Sφ
DΦ
ρDΦ∇φ( )∇• Sφ+ 0=
ρ
Φ
φ Φ ρ⁄=
Basic Solver Capability Theory: Rotational Forces
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• is a volumetric source term, with units of conserved quantity per unit volume per
unit time
• is the kinematic diffusivity for the scalar.
Algebraic Equation
Additional variables may be set up as either algebraic equations or transport equations. For
algebraic additional variables, you must provide an expression for its value throughout the
domain. An algebraic additional variable definition can be as simple as a copy of an existing
variable.
Important: Results obtained using an additional variable that references an existing variablemay differ from results obtained using the referenced variable directly. This is because thecalculation recipes used for such additional variables may differ from those used for thevariable referenced. An example of this occurs when the referenced variable is a wall orboundary only variable. In particular, a variable-specific recipe may exist to gatherintegration point values to nodes and this recipe may differ from the generic recipe appliedfor additional variables.
Rotational Forces
For flows in a rotating frame of reference, rotating at a constant angular velocity ,
additional sources of momentum are required to account for the effects of the Coriolis force
and the centrifugal force:
(Eqn. 139)
where:
(Eqn. 140)
(Eqn. 141)
and where is the location vector and is the relative frame velocity (i.e., the rotating
frame velocity for a rotating frame of reference).
In the energy equation, the advection of total enthalpy is replaced by the advection of
rothalpy, , given by:
(Eqn. 142)
Since the rotation energy is not included in the transient term in the energy equation,
rotational enthalpy is only conserved in the transient solution if is constant.
Sφ
DΦ
ω
SM ,rot SCor Scfg+=
SCor 2ρω U×–=
Scfg ρω ω r×( )×–=
r U
I
I hstat12--U 2 1
2--ω2R2–+=
ω
Basic Solver Capability Theory: Sources
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Alternate Rotation Model
By default, the advection term in the momentum equation models the relative frame
velocity:
(Eqn. 143)
The alternate rotation model modifies the advection term so that it involves the absolute
frame velocity, , instead of the relative frame velocity, . The change of variable in the
advection term requires a modification of the original Coriolis source term. The final form of
the momentum equation becomes:
(Eqn. 144)
Modeling details and recommendations on when this should be used are available. For
details, see Alternate Rotation Model (p. 24 in "ANSYS CFX-Solver Modeling Guide").
Sources
Additional source terms can be applied either to a volume defined by a subdomain, or to a
point within a domain. A point source is actually implemented as a volumetric source within
a single domain element whose center is nearest to the specified point.
For details, see Sources (p. 25 in "ANSYS CFX-Solver Modeling Guide").
Momentum Sources
Momentum sources can be used to model isotropic losses in porous regions, directional
losses in porous regions, or other processes. These situations are described further in the
following sections. More information on using the momentum source models in ANSYS CFX
is available. For details, see General Momentum Source (p. 28 in "ANSYS CFX-Solver
Modeling Guide").
ρU∂t∂
----------- ρU U⊗( )∇•+
pδ– µ ∇U ∇U( )T+( )+( ) 2ρω U× ρω ω r×( )×––∇•=
Uabs U
ρU∂t∂
----------- ρU Uabs⊗( )∇•+
pδ– µ ∇U ∇U( )T+( )+( ) ρω U× ρω ω r×( )×––∇•=
Basic Solver Capability Theory: Sources
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Isotropic LossModel
The momentum loss through an isotropic porous region can be formulated using
permeability and loss coefficients as follows:
(Eqn. 145)
where is the permeability and is the loss coefficient. The linear component of
this source represents viscous losses and the quadratic term represents inertial losses.
The source may alternatively be formulated using linear and quadratic resistance
coefficients, substituting two coefficients and as follows:
(Eqn. 146)
(Eqn. 147)
Directional LossModel
The momentum source through an anisotropic porous region (such as a honeycomb or
perforated plate) may be modeled using the directional loss model. With this model, the
streamwise direction (which is permitted to vary in space), must be specified. Define a local
coordinate system ( ) such that the -axis is aligned with the streamwise direction
and the axes lie on the transverse plane. The momentum losses in these directions are:
(Eqn. 148)
where and are the streamwise and transverse permeabilities, and and
are the streamwise and transverse loss coefficients. These quantities may also be
expressed in terms of linear and quadratic resistance coefficients.
SM x,µ
Kperm--------------U x– K loss
ρ2--- U U x–=
SM y,µ
Kperm--------------U y– K loss
ρ2--- U U y–=
SM z,µ
Kperm--------------U z– K loss
ρ2--- U U z–=
K perm K loss
CR1 CR2
CR1µ
Kperm--------------=
CR2 K lossρ2---=
x' y' z', , x'
y' z',
SM x',µ
KpermS
--------------U x'– K lossS ρ
2--- U U x'–=
SM y',µ
KpermT
--------------U y'– K lossT ρ
2--- U U y'–=
SM z',µ
KpermT
--------------U z'– K lossT ρ
2--- U U z'–=
K permS K perm
T K lossS
K lossT
Basic Solver Capability Theory: Sources
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In addition, the transverse loss may be modeled by multiplying the streamwise coefficients
by some factor. If this option is used together with a streamwise permeability, the implied
transverse permeability is equal to the streamwise permeability divided by this factor. For
details, see Directional Loss Model (p. 27 in "ANSYS CFX-Solver Modeling Guide").
GeneralMomentumSources
The general momentum source is available for specifying momentum sources which are not
covered by the isotropic or directional loss models. A different source can be specified for
each direction as follows:
(Eqn. 149)
(Eqn. 150)
(Eqn. 151)
where the quantities are the specified momentum components. In ANSYS CFX,
Release 11.0, a representative scalar linearization coefficient based on the derivative:
(Eqn. 152)
may be specified to provide robust convergence when a general momentum source is
present.
General Sources
Any scalar equation may have a source term . This covers the energy, additional variable,
turbulence and mass fraction equations.
A linearization coefficient
(Eqn. 153)
may also be specified to provide robust convergence when sources are present. For the
energy equation, the derivative is actually taken with respect to temperature.
For details, see General Sources (p. 26 in "ANSYS CFX-Solver Modeling Guide").
Mass (Continuity) Sources
The mass source contribution, , to the conservation equation for the fluid mass is
specified exactly as for a general source.
SM x, Sspec x, i=
SM y, Sspec y, j=
SM z, Sspec z, k=
Sspec x,
∂Sm
∂U---------
Sφ
∂Sφ∂φ--------
SMS
Basic Solver Capability Theory: Boundary Conditions
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The mass source contributes an additional term to all other transported variables :
(Eqn. 154)
The exterior must be specified in the case that is positive, in which case, the
source behaves very much like an inlet boundary condition.
For details, see Mass (Continuity) Sources (p. 28 in "ANSYS CFX-Solver Modeling Guide").
Bulk Sources
For details, see Sources in Multiphase Flow (p. 170).
Radiation Sources
For details, see Radiation Theory (p. 263).
Boundary Sources
For details, see Boundary Sources (p. 25 in "ANSYS CFX-Solver Modeling Guide").
Boundary Conditions
The following topics will be discussed:
• Inlet (subsonic) (p. 46)
• Inlet (supersonic) (p. 51)
• Outlet (subsonic) (p. 51)
• Outlet (supersonic) (p. 54)
• Opening (p. 55)
Inlet (subsonic)
Mass andMomentum
Normal Speed inThe magnitude of the inlet velocity is specified and the direction is taken to be normal to the
boundary. The direction constraint requires that the flow direction, , is parallel to the
boundary surface normal, which is calculated at each element face on the inlet boundary.
Cartesian Velocity ComponentsThe boundary velocity components are specified, with a non-zero resultant into the domain.
(Eqn. 155)
φ
SMSφ max SMS 0,( )φMS min SMS 0,( )φ+=
φMS SMS
Di
U Inlet U speci V spec j W speck+ +=
Basic Solver Capability Theory: Boundary Conditions
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Cylindrical Velocity ComponentsIn this case the velocity boundary condition is specified in a local cylindrical coordinate
system. Only the axial direction of the local coordinate system needs to be given and the
components of velocity in the r, theta and z directions are automatically transformed by the
ANSYS CFX-Solver into Cartesian velocity components. So, in this case you would specify:
(Eqn. 156)
and the solver will compute the rotation matrix which transforms these components from
the cylindrical components to the Cartesian components such that the boundary condition
is the same as if Cartesian components were specified:
(Eqn. 157)
For details, see Cylindrical Velocity Components (p. 55 in "ANSYS CFX-Solver Modeling
Guide").
Total Pressure
The Total Pressure, , is specified at an inlet boundary condition and the ANSYS
CFX-Solver computes the static pressure needed to properly close the boundary condition.
For rotating frames of reference one usually specifies the stationary frame total pressure
instead.
The direction constraint for the Normal To Boundary option is the same as that for the
Normal Speed In option. Alternatively, the direction vector can be specified explicitly in
terms of its three components. In both cases, the boundary mass flow is an implicit result of
the flow simulation.
Mass Flow RateThe boundary mass flow rate is specified along with a direction component. If the flow
direction is specified as normal to the boundary, a uniform mass influx is assumed to exist
over the entire boundary. Also, if the flow direction is set using Cartesian or cylindrical
components, the component normal to the boundary condition is ignored and, again, a
uniform mass influx is assumed. The mass influx is calculated using:
(Eqn. 158)
where
(Eqn. 159)
U Inlet U r spec, r Uθ spec, θ U z spec, z+ +=
U Inlet U speci V spec j W speck+ +=
ptot
ρU m
AdS∫----------=
AdS∫
Basic Solver Capability Theory: Boundary Conditions
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is the integrated boundary surface area at a given mesh resolution. The area varies with
mesh resolution because the resolution determines how well resolved the boundary
surfaces are. The value of is held constant over the entire boundary surface.
Turbulence For the - turbulence model and Reynolds stress models, the inlet turbulence quantities,
and , are either specified directly or calculated using expressions which scale the
distribution at the inlet according to the turbulence intensity, , where:
(Eqn. 160)
The inlet flows of and involve advection and diffusion.
(Eqn. 161)
(Eqn. 162)
The advection flows are evaluated using the computed inlet values of and :
(Eqn. 163)
(Eqn. 164)
The diffusion flows are assumed to be negligible compared to advection, and are equated
to zero.
Default Intensity and Autocompute Length ScaleWhen default inlet turbulence intensity is selected, the value is set to:
(Eqn. 165)
which is an approximate value for internal pipe flow. The inlet turbulence energy is
calculated using:
(Eqn. 166)
and the turbulence dissipation calculated using:
(Eqn. 167)
ρU
k ε
k ε
I
I uU----=
k ε
QInletk Qadvect
k Qdiffusk
+=
QInletε Qadvect
ε Qdiffusε
+=
k ε
Qadvectk mkspec=
Qadvectε mεspec=
I uU---- 0.037= =
kInlet32--I2U 2
=
εInlet ρCµk2
µt-----=
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where:
(Eqn. 168)
Intensity and Autocompute Length Scale
The turbulence intensity is specified directly and the distributions of and at the inlet
calculated using the same relationships as the Default Intensity and Autocompute
Length Scale option.
Intensity and Length ScaleThe turbulence intensity and length scale are both specified. The turbulence kinetic energy
and dissipation are calculated using:
(Eqn. 169)
and
(Eqn. 170)
k and Epsilon
Both and are specified directly:
(Eqn. 171)
and
(Eqn. 172)
When the Reynolds stress model is employed, the Inlet boundary conditions are specified
with the same turbulence options as those for the - model. Additionally, the stress
tensors are extracted using the computed value of . This is done by assuming the Inlet
boundary to be isotropic with respect to the Reynolds stresses, such that the normal stress
components are:
(Eqn. 173)
and the shear stress components are equal to zero:
(Eqn. 174)
µt 1000Iµ=
k ε
kInlet32--I2U 2
=
εInletk
32--
lt-----=
k ε
kInlet kspec=
εInlet εspec=
k ε
k
uxux uyuy uzuz13--k= = =
uxuy uxuz uyuz 0= = =
Basic Solver Capability Theory: Boundary Conditions
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Heat Transfer Static TemperatureThe inlet static temperature is specified:
(Eqn. 175)
The inlet energy flow involves advection and diffusion.
(Eqn. 176)
The energy flow by advection is a function of the specific total enthalpy, :
(Eqn. 177)
where is computed from the specific static enthalpy, , and the inlet boundary
velocity:
(Eqn. 178)
The static enthalpy is computed using the specified value of , the boundary values of
and , and the thermodynamic relationship for for the given fluid. The
evaluation of depends upon the nature of the mass and momentum specification for the
boundary condition.
The Inlet energy flow by diffusion is assumed to be negligible compared to advection, and
equated to zero.
Total TemperatureThe boundary advection and diffusion terms for specified total temperature are evaluated
in exactly the same way as specified static temperature, except that the static temperature
is dynamically computed from the definition of total temperature:
(Eqn. 179)
which for a fluid with constant heat capacity is:
(Eqn. 180)
Additional information on the treatment of variable specific heat is available. For details, see
Ideal Gas Equation of state (p. 25).
T stat, Inlet T spec=
QInlet Qadvect Qdiffus+=
htot
Qadvect mhtot=
htot hstat
htot hstat12--U 2
+=
T spec
U p h h p T,( )=
U
T tot, Inlet T spec=
T stat, Inlet T tot, InletU 2
2cp--------–=
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AdditionalVariables
The value of the additional variable is specified explicitly at an inlet:
(Eqn. 181)
The inlet flow of involves advection and diffusion:
(Eqn. 182)
and the advection quantity is evaluated using the specified value of :
(Eqn. 183)
The inlet flow by diffusion is assumed to be negligible compared to advection, and set to
zero.
Inlet (supersonic)
Mass andMomentum
At a supersonic Inlet boundary, all quantities must be specified. Static pressure, Cartesian
components of velocity and static or total temperature are required.
Velocity Components and PressureFor a supersonic Inlet boundary, you must specify the components of velocity together with
a value for the relative static pressure.
The velocity components can be Cartesian or cylindrical in the same way as for a subsonic
inlet. For details, see Inlet (subsonic) (p. 46).
Heat Transfer Static TemperatureThe static temperature is specified at the supersonic inlet boundary:
(Eqn. 184)
Total TemperatureTotal temperature is specified at the supersonic inlet boundary:
(Eqn. 185)
Static temperature is dynamically computed from the definition of total temperature.
Outlet (subsonic)
Mass andMomentum
Static Pressure (Uniform)Relative Static Pressure is specified over the outlet boundary:
(Eqn. 186)
ΦInlet Φspec=
Φ
QInletΦ Qadvect
Φ QdiffusΦ
+=
Φ
QadvectΦ mΦspec=
T stat, Inlet T spec=
T tot, Inlet T spec=
pstat, Outlet pspec=
Basic Solver Capability Theory: Boundary Conditions
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Normal SpeedThe magnitude of the outlet velocity is specified and the direction is taken to be normal to
the boundary at mesh resolution.
Cartesian Velocity ComponentsThe boundary velocity components are specified, with a non-zero resultant out of the
domain.
(Eqn. 187)
Cylindrical Velocity ComponentsThese are handled the same way as for an Inlet (subsonic) boundary condition.
Average Static Pressure: OverallThe Outlet Relative Static Pressure is constrained such that the average is the specified
value:
(Eqn. 188)
where the integral is over the entire outlet boundary surface. To enforce this condition,
pressure at each boundary integration point is set as:
(Eqn. 189)
So, the integration point pressure in this case is set to the specified value plus the difference
between the local nodal value and the average outlet boundary pressure. In this way the exit
boundary condition pressure profile can float, but the average value is constrained to the
specified value.
Average Static Pressure: Above or Below Specified RadiusIn this case, the average pressure is only constrained in the region above or below the
specified radius by shifting the calculated pressure profile by the difference between the
specified average and the nodal average above or below the specified radius.
Average Static Pressure: CircumferentialThe circumferential averaging option divides the exit boundary condition into
circumferential bands (oriented radially or axially depending on the geometry). The
pressure within each band is constrained to the specified average pressure value the same
way as is done for the overall averaging:
(Eqn. 190)
where the specified value is applied within a band and the nodal average pressure is also
calculated within a band.
UOutlet U speci V spec j W speck+ +=
pspec1A---- pn Ad
S∫=
pip pspec pnode pnode–( )+=
pip pspec ηk( ) pn pn ηk( )–( )+=
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Mass Flow Rate: Scale Mass FlowsThe mass flux distribution across the outlet is determined by starting with the local mass
flow rate distribution calculated by the flow solver at each integration point:
(Eqn. 191)
From that distribution, you calculate the estimated total mass flow rate through the outlet
boundary condition:
(Eqn. 192)
where the summation is over all boundary integration points. A scaling factor is computed
at the end of each coefficient loop which is used to scale the local integration point mass
flows such that they add up to the specified mass flow rate:
(Eqn. 193)
Iteratively, during the computation, can be greater than or less than unity. The final
integration point mass flows are reset by multiplying the integration point mass flows by
the scaling factor:
(Eqn. 194)
In this way, the mass flux profile is an implicit result of the solution and at the same time
gives exactly the specified mass flow rate.
Mass Flow Rate: Shift Pressure with Pressure ProfileThis condition differs from the last one in that pressure is shifted in the continuity equation
to get the specified mass flow rate. Generally speaking, the mass flow rate at each boundary
integration point is dependent upon both velocity and pressure:
(Eqn. 195)
where the integration point velocity depends upon nodal velocity and integration point
pressures through the Rhie-Chow coupling. For this boundary condition, the integration
point pressures are given by an expression of the form:
(Eqn. 196)
mip˙ ρipAipU ip=
mtotest mip
˙all∑=
Fmspec
mtotest
------------=
F
mip˙ FρipAipU ip=
mip˙ ρipAipU ip pip U node,( )=
pip F pprof 1 F–( ) pnode pnode–( ) pshift+ +=
Basic Solver Capability Theory: Boundary Conditions
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where is an optional specified pressure profile, is the boundary node pressure,
is the outlet boundary patch nodal average pressure, is the profile blend factor
which sets how much the specified profile influences the boundary condition, and is
the level shift factor automatically computed by the ANSYS CFX-Solver each coefficient loop
to enforce the specified mass flow rate, such that:
(Eqn. 197)
where the sum, in this case, is over all the outlet boundary condition integration points.
Mass Flow Rate: Shift Pressure with Circumferential Pressure AveragingA further extension of the shift pressure feature for outlet mass flow rate conditions (or
outlet boundaries using an Average Static Pressure specification) enforces the specified
profile as an average pressure profile (or average pressure) in circumferential bands (radial
or axial), held at a particular value.
Starting with the original formula for the integration point pressure in (Eqn. 196): instead of
imposing a particular profile distribution, an average pressure profile within bands is
introduced:
(Eqn. 198)
where is the average pressure desired in band , and is the current
average nodal value in band . corresponds to the pressure profile blend. When the
specified profile spatially varies, the flow solver will compute the average of that profile
within each band and then use those values for .
Turbulence,Heat Transfer,ConvectedAdditionalVariables andOther Scalars
For scalar quantities, the ANSYS CFX-Solver imposes a constant gradient constraint
(generally non-zero) at the outlet boundary.
Outlet (supersonic)
The specification of a supersonic outlet boundary condition requires no accompanying
values, except for the radiation intensity if radiation is modeled.
pprof pnode
pnode F
pshift
mspec ρip∑ ipAipU ip pip U node,( )=
pip F pprof ηk( ) pnode pnode– ηk( )( ) pshift+ +=
pprof ηk( ) ηk pnode ηk( )
ηk F
pprof ηk( )
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Opening
An opening boundary condition allows the fluid to cross the boundary surface in either
direction. For example, all of the fluid might flow into the domain at the opening, or all of
the fluid might flow out of the domain, or a mixture of the two might occur. An opening
boundary condition might be used where it is known that the fluid flows in both directions
across the boundary.
Mass andMomentum
Cartesian Velocity ComponentsThe Cartesian components of the flow velocity are specified at the opening boundary:
(Eqn. 199)
Cylindrical Velocity ComponentsThese are handled the same way as for an Inlet (subsonic) boundary condition.
Pressure and DirectionAn opening boundary condition can also be specified with a Relative Pressure value:
(Eqn. 200)
The value is interpreted as relative total pressure for inflow and relative static pressure for
outflow.
The direction component, normal to boundary condition or direction components, must
also be specified. The magnitude of the velocity at the opening boundary is then part of the
solution.
You can optionally specify a loss coefficient, :
(Eqn. 201)
with a velocity component specification at an opening boundary. The pressure drop is
calculated using:
(Eqn. 202)
where is the magnitude of the velocity component normal to the opening boundary.
For inflows, the constraint on pressure and velocity becomes:
(Eqn. 203)
UOpening U speci V spec j W speck+ +=
pOpening pspec=
f
f f spec=
∆ ploss12-- fρU n
2=
U n
pspec12-- fρU n
2– pstat=
Basic Solver Capability Theory: Boundary Conditions
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and for outflows, the constraint becomes:
(Eqn. 204)
The coefficient can be used to model the pressure drop across a screen or other planar
resistance.
Heat Transfer Static TemperatureStatic temperature is specified at the opening boundary:
(Eqn. 205)
AdditionalVariables
For flow into the domain, the value of the additional variable at an opening boundary is that
specified on the Additional Variable Details tab:
(Eqn. 206)
For flow out of the domain the specified value is not used. The value for flow out of the
domain is calculated by the ANSYS CFX-Solver and is the additional variable value taken
from the solution field.
Wall
Mass andMomentum
No Slip (Not Moving, no Wall Velocity)The velocity of the fluid at the wall boundary is set to zero, so the boundary condition for the
velocity becomes:
(Eqn. 207)
Free SlipIn this case, the velocity component parallel to the wall has a finite value (which is
computed), but the velocity normal to the wall, and the wall shear stress, are both set to
zero:
(Eqn. 208)
(Eqn. 209)
No Slip (Moving, with Wall Velocity)In this case, the fluid at the wall boundary moves at the same velocity as the wall. There are
three different options for the wall velocity:
pspec12-- fρU n
2+ pstat=
T stat,Opening T spec=
ΦOpening Φspec=
UWall 0=
Un,Wall 0=
τw 0=
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• Cartesian Components: You can directly specify Cartesian components in a localcoordinate frame or the global coordinate frame:
(Eqn. 210)
• Cylindrical Components: You can directly specify cylindrical components in a localcylindrical coordinate system:
(Eqn. 211)
and the solver automatically transforms the cylindrical velocity components into theglobal Cartesian coordinate system.
• Rotating Wall: A counter rotating wall can be specified for walls in rotating frames. Inthis case, the wall is stationary in the absolute frame; in the relative frame, it moves witha velocity:
(Eqn. 212)
where is the radial vector from the domain axis of rotation to the wall and is the
domain angular velocity.
A rotating wall can be specified in both stationary and rotating frames. This option isuseful to use in stationary domains when you would like to create a spinning wall. In thiscase, you enter a local rotation axis for the wall boundary and the wall velocity:
(Eqn. 213)
The solver automatically transforms the specified wall velocity into Cartesian components.
This option could be used to duplicate the counter rotating wall option in rotating frames
by explicitly setting the angular velocity equal to minus one times the domain angular
velocity.
Turbulence The treatment of wall boundary conditions for turbulent flow is the same as for laminar flow,
except for No Slip. For details, see Modeling Flow Near the Wall (p. 107).
Heat Transfer AdiabaticThe Adiabatic Wall boundary condition allows no heat transfer across the Wall boundary:
(Eqn. 214)
Fixed TemperatureStatic Temperature is specified at the wall boundary:
(Eqn. 215)
UWall U speci V spec j W speck+ +=
UWall U r,spec r Uθ spec, θ U z,specz+ +=
UWall ωR–=
R ω
UWall ωR=
qw 0=
T stat,w T spec=
Basic Solver Capability Theory: Automatic Time Scale Calculation
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Heat FluxHeat flux at the wall boundary is specified:
(Eqn. 216)
Heat Transfer CoefficientHeat flux at the wall boundary is calculated using:
(Eqn. 217)
where is a specified heat transfer coefficient, is the specified boundary temperature,
(i.e., outside the fluid domain) and is the temperature at the internal near-wall
boundary element center node. For details, see Wall Heat Transfer (p. 72 in "ANSYS
CFX-Solver Modeling Guide").
AdditionalVariables
The options for specifying additional variable quantities at wall boundaries are analogous
to those for heat transfer.
Symmetry Plane
The symmetry plane boundary condition imposes constraints which ‘mirror’ the flow on
either side of it.
For example, the normal velocity component at the symmetry plane boundary is set to zero:
(Eqn. 218)
and the scalar variable gradients normal to the boundary are also set to zero:
(Eqn. 219)
Automatic Time Scale Calculation
This section describes the way in which a timestep is calculated when using the Auto
Timescale or Auto Timescalewith a Maximum Timescale option for setting the timestep
used during the calculation of a solution. For details, see Timestep Selection (p. 374 in
"ANSYS CFX-Solver Modeling Guide").
qw qspec=
qw hc Tb Tnw–( )=
hc Tb
Tnw
Un 0=
n∂∂φ 0=
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Fluid Time Scale Estimate
The following length scales are calculated by ANSYS CFX:
(Eqn. 220)
where is the domain volume (over all domains), , and are the , and
extents of the domain (over all domains) and is the area of an ‘open’ boundary (i.e.,
inlets, outlets or openings).
Further length scales are then calculated depending on the Length Scale Option
parameter, which can be Conservative, Aggressive or Specified Length Scale:
(Eqn. 221)
The velocity scales used to calculate a timestep are:
(Eqn. 222)
where is the arithmetic average of the velocity on a boundary, is the arithmetic
average of the nodal velocities, and are the maximum and minimum
pressure values on an ‘open’ boundary and is the arithmetic average nodal density.
For compressible flows, a mach number for the simulation is calculated as:
(Eqn. 223)
where is the arithmetic averaged speed of sound over all nodes.
Lvol V3=
Lext max Lx Ly Lz, ,( )=
Lbc min Abc=
V Lx Ly Lz x y z
Abc
Lscale
min Lvol Lext( , ) for Conservative
max Lvol Lext( , ) for Aggressive
Luser for Specified⎩⎪⎨⎪⎧
=
U bc max Ubc=
U node Unode=
U∆Ppbc,max pbc,min–
ρnode--------------------------------------=
U bc U node
pbc,max pbc,min
ρnode
Mmax U bc U node U ∆P, ,( )
c-------------------------------------------------------=
c ∂ρ∂p------⎝ ⎠
⎛ ⎞1–
=
c
Basic Solver Capability Theory: Automatic Time Scale Calculation
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For buoyant flows using the full buoyancy model:
(Eqn. 224)
and for the Boussinesq model:
(Eqn. 225)
where is the thermal expansivity and and are the maximum and minimum
domain temperatures (over all domains).
The final fluid time scale used is calculated as:
(Eqn. 226)
where:
(Eqn. 227)
(Eqn. 228)
If , then
(Eqn. 229)
Solid Time Scale Estimate
The solid time scale estimate will, in general, be much larger than the fluid time scale.
The length scale used to calculate a solid time scale, , is calculated in the same manner
as for fluid time scales. The default value is taken as the cube root of the volume of the solid
domain.
A volume averaged diffusion is calculated as:
(Eqn. 230)
g g=
g g β Tmax Tmin–( )=
β Tmax Tmin
∆t min ∆tU ∆t∆P ∆t g ∆trot ∆tc, , , ,( )=
∆tU 0.3Lscale
max U bc U node,( )-----------------------------------------= ∆t∆P 0.3
LscaleU∆P------------=
∆t gLscale
g------------= ∆trot
0.1ω-------=
M 0.3>
∆tcLbc
max U bc U node U∆P c, , ,( )------------------------------------------------------------=
Lscale
α λρcp--------=
Basic Solver Capability Theory: Mesh Adaption
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where each quantity is volume averaged over the solid. The solid time scale is finally
calculated as:
(Eqn. 231)
where is the specified solid timescale factor, which has a default value of 1.
Mesh Adaption
Mesh adaption in ANSYS CFX is the process in which, once or more during a run, the mesh
is selectively refined in areas which depend on the adaption criteria specified. This means
that as the solution is calculated, the mesh can automatically be refined in locations where
solution variables are changing most rapidly, in order to resolve the features of the flow in
these regions.
More information on the mesh adaption process and controls is available. For details, see
Mesh Adaption (p. 205 in "ANSYS CFX-Pre User's Guide"). Within the mesh adaption step
itself, three processes take place:
1. Adaption criteria are calculated for each mesh element.
2. The appropriate number of nodes are added to the existing mesh according to theadaption criteria calculated.
3. The solution already calculated on the older mesh is linearly interpolated onto the new
mesh.
This section describes the details of exactly which elements are refined and how the
refinement takes place.
Adaption Criteria
This section describes how the adaption criteria are calculated for each mesh edge.
• If the adaption criteria method is Solution Variation, then the adaption criteria, , for
a given mesh edge of length is calculated as:
(Eqn. 232)
where is the adaption variable (e.g., density, pressure, etc.), is the global
range of the variable over all the nodes (excluding those on wall boundary
conditions for turbulent flow), is the difference between at one end of the edge
and the other end, and is a scalar for adaption variable to scale all the to take
values between 0 and 1.
∆t fLscale
2
α------------=
f
Ai
i li
Aiφ ji∆
Nφ jφ j∆
--------------------j
∑=
φ j jth ∆φ j
φ j
∆φij φ j
Nφ jj Ai
Basic Solver Capability Theory: Mesh Adaption
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• If the adaption criteria method is Variation * Edge Length, then the adaption criteria
for a given mesh edge of length is calculated as:
(Eqn. 233)
where is the global range of the adaption variable over all the nodes
(excluding those on wall boundary conditions for turbulent flow), is the difference
between at one end of the edge and the other end, and is a length chosen to
scale all the to take values between 0 and 1.
• If you select more than one solution variable, then the adaption criteria are calculatedfor each variable at each edge, and the sum over all adaption variables is used.
• If you specify the adaption criteria to be an expression, then the expression is evaluated
at each node, and the calculation of the adaption criteria follows as if this were another
solution variable.
The edges which have the largest adaption criteria are marked for refinement, providing
that this would not result in mesh edges which were shorter than the Minimum Edge
Length, if this was specified.
Mesh Refinement Implementation in ANSYS CFX
There are two general methods for performing mesh adaption. Incremental adaption
takes an existing mesh and modifies it to meet the adaption criteria. The alternative is
re-meshing, in which the whole geometry is re-meshed at every adaption step according
to the adaption criteria. In ANSYS CFX, incremental adaption is used, since this is much
faster, however, this imposes the limitation that the resulting mesh quality is limited by the
quality of the initial mesh.
The particular implementation of incremental adaption that is adopted in ANSYS CFX is
known as hierarchical refinement or h-refinement. Each adaption step consists of a
structured refinement of an existing mesh. A sequence of refinements form a set of
hierarchical levels.
In each mesh adaption step, each mesh edge which is marked for adaption (see the previous
section) has an extra node placed half-way along it. The mesh elements which share this
edge are then divided to use the new node, subject to the following:
• Neighboring elements must only differ by one refinement level. Hence, one meshelement cannot be divided twice if its neighbor has not been divided at all.
• Where possible, regular refinement of an element takes place. Regular refinementmeans that all the edges of an element are divided into two, and the element splitaccordingly. To make this possible, extra nodes may be added.
• No “hanging” nodes are allowed. This means that if an extra node is added to an edge,all the mesh elements which share that edge must be refined.
Ai i li
Aili φ ji∆
Nφ jφ j∆
--------------------j
∑=
φ j∆ jth φ j
φ ji∆
φ j Nφ j
Ai
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• Only certain types of elements are allowed in the refined mesh: tetrahedron, prism,
pyramid and hexahedron.
Adaption ininflated regionsof the mesh
In regions where inflation has taken place (so that there are prisms and some pyramid
elements near wall boundary conditions), the mesh adaption avoids refining these
elements in the direction perpendicular to the wall. Only edges on the interface between
the inflated elements and the rest of the tetrahedral mesh are allowed to be marked for
adaption. When the refinement of these edges takes place, the refinement propagates
through the layers of prismatic elements to the wall boundary condition itself.
Adaption to theoriginalgeometry
Mesh adaption in ANSYS CFX has the capability of refining the surface mesh back to the
original geometry. This means that nodes which are added to the surface of the problem are
added onto the surfaces of the original geometry rather than to the edges of the existing
mesh. An example is shown below:
Layer of prismaticelements produced as aresult of inflation.
This sort of refinement isnot allowed.
This is the only way thatrefinement takes place in theinflated layers.
Adapt to GeometryDon’t adapt toGeometry
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When using Adapt to Geometry, the adaption process can help to resolve fine features in
the geometry. When this is done, nodes near the surface may be moved slightly in order to
make sure that the quality of the resulting elements is not degraded (not shown in the figure
above).
Note: Since this feature requires the underlying geometry file, it can only be used with GTMfiles and ANSYS CFX databases from previous releases of ANSYS CFX. This limitation will beremoved in future releases.
Mesh Adaption Limitations
Mesh Adaption in ANSYS CFX, Release 11.0 is subject to the following limitations:
• Mesh adaption cannot be used in multidomain simulations or in cases with externalsolver coupling. Mesh adaption also cannot be used for transient, mesh-motion,radiative-tracking, or particle-transport cases.
• The mesh may not be coarsened such that original mesh elements are combined. Meshelements that have been refined in earlier adaption steps may be coarsened, back totheir original mesh elements, but not beyond.
• You must specify how much adaption takes place by specifying the maximum numberof nodes in the final mesh. You cannot specify that mesh adaption should take placeuntil the mesh stops changing or until a particular solution criterion is satisfied.
• The method used is not well-suited to meshes with many high-aspect ratio elements,since it can only refine elements in an isotropic manner.
• The method cannot improve the quality of the existing mesh.
Mesh AdaptionTips
• More than one adaption step applied to a surface mesh with no underlying geometrycan lead to convergence problems with transonic problems.
• If you set Mesh Adaption Criteria Method toSolution Variationwithout specifyingMinimum Edge Length, then you may over-refine in regions where there arediscontinuities. If you wish to adapt meshes containing geometrically small features,you can exploit the adaption capability without limitation by not setting any minimumedge length.
• If you set Mesh Adaption Criteria Method to Variation * Edge Length, then youwill not over-refine discontinuities. This method places an emphasis on adapting longedges with large variations of solution variable in preference to short edges with largevariations of solution variable. This method will also refine long edges with a smallvariation of solution variable.
• Adaption criteria values for each node will be computed as the average value of alledges connected to the node, and these values will be stored in the results file forvisualization purposes.
• If you find that the mesh adaption appears to have missed a discontinuity in a solution
by refining the mesh in front of or behind the discontinuity, then your solution was not
sufficiently converged before mesh adaption took place. As a solution containing a
discontinuity develops, the location of such a discontinuity may move. If the solution is
not sufficiently converged before mesh adaption takes place, then the mesh will be
Basic Solver Capability Theory: Flow in Porous Media
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refined at the location of the discontinuity, which will move as convergence continues.
A lower Target Residual can be set on the Mesh Adaption/Advanced Parameters tab
to correct this problem.
Flow in Porous Media
Flow in porous media in ANSYS CFX can be calculated using either a model for momentum
loss or a full porous model. The momentum loss model is available in fluid domains, while
the full porous loss model is only available in porous domains.
Darcy Model
The porous model is at once both a generalization of the Navier-Stokes equations and of
Darcy's law commonly used for flows in porous regions. It can be used to model flows where
the geometry is too complex to resolve with a grid. The model retains both advection and
diffusion terms and can therefore be used for flows in rod or tube bundles where such
effects are important.
In deriving the continuum equations, it is assumed that ‘infinitesimal’ control volumes and
surfaces are large relative to the interstitial spacing of the porous medium, though small
relative to the scales that you wish to resolve. Thus, given control cells and control surfaces
are assumed to contain both solid and fluid regions.
The volume porosity at a point is the ratio of the volume available to flow in an
infinitesimal control cell surrounding the point, and the physical volume of the cell.
Hence:
(Eqn. 234)
It is assumed that the vector area available to flow, , through an infinitesimal planar
control surface of vector area is given by:
(Eqn. 235)
where is a symmetric second rank tensor, called the area porosity tensor.
Recall that the dot product of a symmetric rank two tensor with a vector is the vector.
ANSYS CFX presently only allows to be isotropic.
The general scalar advection-diffusion equation in a porous medium becomes:
(Eqn. 236)
γ V '
V
V ' γV=
A'
A
A' K A⋅=
K K ij( )=
K Ai⋅ K ijA j= K
t∂∂ γρΦ( ) ρK U⋅ Φ( )∇• ΓK Φ∇⋅( )∇•–+ γS=
Basic Solver Capability Theory: Flow in Porous Media
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In addition to the usual production and dissipation terms, the source term will contain
transfer terms from the fluid to the solid parts of the porous medium.
In particular, the equations for conservation of mass and momentum are:
(Eqn. 237)
and:
(Eqn. 238)
(Eqn. 239)
where is the true velocity, is the effective viscosity - either the laminar viscosity or a
turbulent quantity, and represents a resistance to flow in the porous medium.
This is in general a symmetric positive definite second rank tensor, in order to account for
possible anisotropies in the resistance.
In the limit of large resistance, a large adverse pressure gradient must be set up to balance
the resistance. In that limit, the two terms on the r.h.s. of (Eqn. 239) are both large and of
opposite sign, and the convective and diffusive terms on the l.h.s. are negligible. Hence,
(Eqn. 239) reduces to:
(Eqn. 240)
Hence, in the limit of large resistance, you obtain an anisotropic version of Darcy's law, with
permeability proportional to the inverse of the resistance tensor. However, unlike Darcy's
law, you are working with the actual fluid velocity components , which are discontinuous
at discontinuity in porosity, rather than the continuous averaged superficial velocity,
Heat transfer can be modeled with an equation of similar form:
(Eqn. 241)
where is an effective thermal diffusivity and contains a heat source or sink to or from
the porous medium.
S
t∂∂ γρ ρK U⋅( )∇•+ 0=
t∂∂ γρU( ) ρ K U⋅( ) U⊗( )∇• µeK U∇ U∇( )T
+( )⋅( )∇•–+
γR U⋅– γ p∇–=
U µe
R Rij( )=
U R 1– p∇⋅–=
U
Q K U⋅=
t∂∂ γρH( ) ρK U⋅ H( )∇• ΓeK H∇⋅( )∇•–+ γ SH
=
Γe SH
Basic Solver Capability Theory: Flow in Porous Media
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Directional Loss Model
From the general momentum equation for a fluid domain:
(Eqn. 242)
the momentum source, can be represented by:
(Eqn. 243)
where:
• is a linear resistance coefficient
• is a quadratic resistance coefficient
• contains other momentum sources (which may be directional)
• and are superficial velocities
Darcy’s Law A generalized form of Darcy’s law is given by:
(Eqn. 244)
where:
• is the dynamic viscosity
• is the permeability
• is the empirical loss coefficient
Implementationin ANSYS CFX
Comparing (Eqn. 243) with (Eqn. 244), the following coefficients are set:
(Eqn. 245)
Data may sometimes be expressed in terms of the true velocity, whereas ANSYS CFX uses
superficial velocity. If so, the coefficients are represented by:
(Eqn. 246)
where is the porosity.
∂ ρU i( )∂t
-----------------∂ ρU jU i( )
∂x j------------------------+ ∂p
∂xi-------– ρgi
∂τ ji∂x j--------- Si
M+ + +=
SiM
SiM CR1U i– CR2 U U i– Si
spec+=
CR1
CR2
Sispec
U U
∂p∂xi-------–
µK perm--------------U i K loss
ρ2--- U U i+=
µ
K perm
K loss
CR1 µK---- CR2, K loss
ρ2---= =
CR1 µγK------- CR2,
K lossρ
2γ 2--------------= =
γ
Basic Solver Capability Theory: Flow in Porous Media
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ANSYS CFX-Solver Theory Guide
Turbulence and Wall FunctionTheory
Introduction
The topics in this section include:
• Turbulence Models (p. 69)
• Eddy Viscosity Turbulence Models (p. 72)
• Reynolds Stress Turbulence Models (p. 85)
• ANSYS CFX Transition Model Formulation (p. 92)
• Large Eddy Simulation Theory (p. 97)
• Detached Eddy Simulation Theory (p. 100)
• Modeling Flow Near the Wall (p. 107)
• Wall Distance Formulation (p. 117)
Turbulence Models
Turbulence consists of fluctuations in the flow field in time and space. It is a complex
process, mainly because it is three dimensional, unsteady and consists of many scales. It can
have a significant effect on the characteristics of the flow. Turbulence occurs when the
inertia forces in the fluid become significant compared to viscous forces, and is
characterized by a high Reynolds Number.
In principle, the Navier-Stokes equations describe both laminar and turbulent flows without
the need for additional information. However, turbulent flows at realistic Reynolds numbers
span a large range of turbulent length and time scales, and would generally involve length
scales much smaller than the smallest finite volume mesh, which can be practically used in
a numerical analysis. The Direct Numerical Simulation (DNS) of these flows would require
computing power which is many orders of magnitude higher than available in the
foreseeable future.
Turbulence and Wall Function Theory: Turbulence Models
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To enable the effects of turbulence to be predicted, a large amount of CFD research has
concentrated on methods which make use of turbulence models. Turbulence models have
been specifically developed to account for the effects of turbulence without recourse to a
prohibitively fine mesh and direct numerical simulation. Most turbulence models are
statistical turbulence model, as described below. The two exceptions to this in ANSYS CFX
are the Large Eddy Simulation model and the Detached Eddy Simulation model.
• Large Eddy Simulation Theory (p. 97)
• Detached Eddy Simulation Theory (p. 100)
Statistical Turbulence Models and the Closure Problem
When looking at time scales much larger than the time scales of turbulent fluctuations,
turbulent flow could be said to exhibit average characteristics, with an additional
time-varying, fluctuating component. For example, a velocity component may be divided
into an average component, and a time varying component.
In general, turbulence models seek to modify the original unsteady Navier-Stokes equations
by the introduction of averaged and fluctuating quantities to produce the Reynolds
Averaged Navier-Stokes (RANS) equations. These equations represent the mean flow
quantities only, while modeling turbulence effects without a need for the resolution of the
turbulent fluctuations. All scales of the turbulence field are being modeled. Turbulence
models based on the RANS equations are known as Statistical Turbulence Models due to the
statistical averaging procedure employed to obtain the equations.
Simulation of the RANS equations greatly reduces the computational effort compared to a
Direct Numerical Simulation and is generally adopted for practical engineering calculations.
However, the averaging procedure introduces additional unknown terms containing
products of the fluctuating quantities, which act like additional stresses in the fluid. These
terms, called ‘turbulent’ or ‘Reynolds’ stresses, are difficult to determine directly and so
become further unknowns.
The Reynolds (turbulent) stresses need to be modeled by additional equations of known
quantities in order to achieve “closure.” Closure implies that there is a sufficient number of
equations for all the unknowns, including the Reynolds-Stress tensor resulting from the
averaging procedure. The equations used to close the system define the type of turbulence
model.
ReynoldsAveragedNavier-Stokes(RANS)Equations
As described above, turbulence models seek to solve a modified set of transport equations
by introducing averaged and fluctuating components. For example, a velocity may be
divided into an average component, , and a time varying component, .
(Eqn. 1)
U
U u
U U u+=
Turbulence and Wall Function Theory: Turbulence Models
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The averaged component is given by:
(Eqn. 2)
where is a time scale that is large relative to the turbulent fluctuations, but small relative
to the time scale to which the equations are solved. For compressible flows, the averaging
is actually weighted by density (Favre-averaging), but for simplicity, the following
presentation assumes that density fluctuations are negligible.
For transient flows, the equations are ensemble-averaged. This allows the averaged
equations to be solved for transient simulations as well. The resulting equations are
sometimes called URANS (Unsteady Reynolds Averaged Navier-Stokes equations).
Substituting the averaged quantities into the original transport equations results in the
Reynolds-averaged equations given below. For details, see Transport Equations (p. 23). In
the following equations, the bar is dropped for averaged quantities, except for products of
fluctuating quantities.
(Eqn. 3)
(Eqn. 4)
where is the molecular stress tensor.
The continuity equation has not been altered but the momentum and scalar transport
equations contain turbulent flux terms additional to the molecular diffusive fluxes. These
are the Reynolds stress, , and the Reynolds flux, . These terms arise from the
non-linear convective term in the un-averaged equations. They reflect the fact that
convective transport due to turbulent velocity fluctuations will act to enhance mixing over
and above that caused by thermal fluctuations at the molecular level. At high Reynolds
numbers, turbulent velocity fluctuations occur over a length scale much larger than the
mean free path of thermal fluctuations, so that the turbulent fluxes are much larger than the
molecular fluxes.
The Reynolds-averaged energy equation is:
(Eqn. 5)
U 1∆t------ U td
t
t ∆t+
∫=
∆t
ρ∂t∂
------ ρU( )∇•+ 0=
ρU∂t∂
------------ ρU U⊗ ∇•+ τ ρ– u u⊗ SM+∇•=
τ
ρu u⊗ ρuφ
ρhtot∂t∂
-------------- p∂t∂
------– ρUhtot( )∇•+ λ∇τ ρuh–( ) ∇ U τ•( ) SE+•+∇•=
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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This equation contains an additional turbulence flux term, compared with the
instantaneous equation. For details on this, see Equation 80. The term in the
equation is the viscous work term that can be included by enabling Viscous Work in ANSYS
CFX-Pre.
The mean Total Enthalpy is given by:
(Eqn. 6)
Note that the Total Enthalpy contains a contribution from the turbulent kinetic energy, k,
given by:
(Eqn. 7)
Similarly, the additional variable equation becomes
(Eqn. 8)
Turbulence models close the Reynolds-averaged equations by providing models for the
computation of the Reynolds stresses and Reynolds fluxes. ANSYS CFX models can be
broadly divided into two classes: eddy viscosity models and Reynolds stress models.
Eddy Viscosity Turbulence Models
One proposal suggests that turbulence consists of small eddies which are continuously
forming and dissipating, and in which the Reynolds stresses are assumed to be proportional
to mean velocity gradients. This defines an ‘eddy viscosity model.’
The eddy viscosity hypothesis assumes that the Reynolds stresses can be related to the
mean velocity gradients and Eddy (turbulent) Viscosity by the gradient diffusion hypothesis,
in a manner analogous to the relationship between the stress and strain tensors in laminar
Newtonian flow:
(Eqn. 9)
where is the Eddy Viscosity or Turbulent Viscosity. This has to be modeled.
ρuh
∇ U τ•( )•
htot h 12--U 2 k+ +=
k 12--u2
=
ρφ∂t∂
--------- ρUφ( )∇•+ Γ∇φ ρ– uφ( ) Sφ+∇•=
ρu u⊗– µt ∇U ∇U( )T+( ) 2
3--δ ρk µt∇ U⋅+( )–=
µt
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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Analogous to the eddy viscosity hypothesis is the eddy diffusivity hypothesis, which states
that the Reynolds fluxes of a scalar are linearly related to the mean scalar gradient:
(Eqn. 10)
where is the Eddy Diffusivity, and this has to be prescribed. The Eddy Diffusivity can be
written as:
(Eqn. 11)
where is the turbulent Prandtl number. Eddy diffusivities are then prescribed using the
turbulent Prandtl number.
The above equations can only express the turbulent fluctuation terms of functions of the
mean variables if the turbulent viscosity, , is known. Both the - and - two-equation
turbulence models provide this variable.
Subject to these hypotheses, the Reynolds averaged momentum and scalar transport
equations become:
(Eqn. 12)
where is the sum of the body forces, and is the Effective Viscosity defined by:
(Eqn. 13)
and is a modified pressure, defined by:
(Eqn. 14)
By default, the solver actually assumes that , but the contribution can be
activated by setting the expert parameter “pressure value option = 1“. In Equation 12 above,
there is a term which although included in the fundamental form of the
equation (For details, see The Momentum Equations (p. 23 in "ANSYS CFX-Solver Theory
Guide").) is neglected in the solver and thus not included here.
ρuφ– Γt∇φ=
Γt
ΓtµtPrt-------=
Prt
µt k ε k ω
ρU∂t∂
------------ ρU U⊗( )∇•+ B ∇ p'– µeff ∇U ∇U( )T+( )( )∇•+=
B µeff
µeff µ µt+=
p'
p' p 23--ρk 2
3--µt∇ U⋅+ +=
p' p=23--ρk
u 23--δ∇ U•⎝ ⎠
⎛ ⎞–
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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The Reynolds averaged energy equation becomes:
(Eqn. 15)
Note that although the transformation of the molecular diffusion term may be inexact if
enthalpy depends on variables other than temperature, the turbulent diffusion term is
correct, subject to the eddy diffusivity hypothesis. Moreover, as turbulent diffusion is usually
much larger than molecular diffusion, small errors in the latter can be ignored.
Similarly, the Reynolds averaged transport equation for Additional Variables (non-reacting
scalars) becomes:
(Eqn. 16)
Eddy viscosity models are distinguished by the manner in which they prescribe the eddy
viscosity and eddy diffusivity.
The Zero Equation Model in ANSYS CFX
Very simple eddy viscosity models compute a global value for from the mean velocity
and a geometric length scale using an empirical formula. Because no additional transport
equations are solved, these models are termed ‘zero equation.’
The zero equation model in ANSYS CFX uses an algebraic equation to calculate the viscous
contribution from turbulent eddies. A constant turbulent eddy viscosity is calculated for the
entire flow domain.
The turbulence viscosity is modeled as the product of a turbulent velocity scale, , and a
turbulence length scale, , as proposed by Prandtl and Kolmogorov,
(Eqn. 17)
where is a proportionality constant. The velocity scale is taken to be the maximum
velocity in the fluid domain. The length scale is derived using the formula:
(Eqn. 18)
where is the fluid domain volume. This model has little physical foundation and is not
recommended.
ρhtot( )∂t∂
------------------- p∂t∂
------ ρUhtot( )∇•+– λ τµtPrt------- h∇+∇⎝ ⎠
⎛ ⎞ ∇ U τ•( ) SE+•+∇•=
ρφ∂t∂
--------- ρUφ( )∇•+ Γφµtσφ------+⎝ ⎠
⎛ ⎞ ∇φ∇• Sφ+=
µt
U t
lt
µt ρ f µU tlt=
f µ
lt V D
13--
⎝ ⎠⎜ ⎟⎛ ⎞
7⁄=
V D
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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Two Equation Turbulence Models
Two-equation turbulence models are very widely used, as they offer a good compromise
between numerical effort and computational accuracy. Two-equation models are much
more sophisticated than the zero equation models. Both the velocity and length scale are
solved using separate transport equations (hence the term ‘two-equation’).
The - and - two-equation models use the gradient diffusion hypothesis to relate the
Reynolds stresses to the mean velocity gradients and the turbulent viscosity. The turbulent
viscosity is modeled as the product of a turbulent velocity and turbulent length scale.
In two-equation models, the turbulence velocity scale is computed from the turbulent
kinetic energy, which is provided from the solution of its transport equation. The turbulent
length scale is estimated from two properties of the turbulence field, usually the turbulent
kinetic energy and its dissipation rate. The dissipation rate of the turbulent kinetic energy is
provided from the solution of its transport equation.
The k-epsilonmodel in ANSYSCFX
k is the turbulence kinetic energy and is defined as the variance of the fluctuations in
velocity. It has dimensions of (L2 T-2); for example, m2/s2. is the turbulence eddy
dissipation (the rate at which the velocity fluctuations dissipate), and has dimensions of
per unit time (L2 T-3); for example, m2/s3.
The - model introduces two new variables into the system of equations. The continuity
equation is then:
(Eqn. 19)
and the momentum equation becomes:
(Eqn. 20)
where is the sum of body forces, is the effective viscosity accounting for turbulence,
and is the modified pressure as defined in Eqn. 15 above.
The - model, like the zero equation model, is based on the eddy viscosity concept, so
that:
(Eqn. 21)
k ε k ω
ε
k
k ε
ρ∂t∂
------ ρU( )∇•+ 0=
ρU∂t∂
------------ ρU U⊗( ) µeff U∇( )∇•–∇•+ p'∇– µeff U∇( )T∇• B+ +=
B µeff
p'
k ε
µeff µ µt+=
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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where is the turbulence viscosity. The - model assumes that the turbulence viscosity
is linked to the turbulence kinetic energy and dissipation via the relation:
(Eqn. 22)
where is a constant. For details, see List of Symbols (p. 2).
The values of and come directly from the differential transport equations for the
turbulence kinetic energy and turbulence dissipation rate:
(Eqn. 23)
(Eqn. 24)
where , , and are constants. For details, see List of Symbols (p. 2).
is the turbulence production due to viscous and buoyancy forces, which is modeled
using:
(Eqn. 25)
For incompressible flow, is small and the second term on the right side of (Eqn. 25)
does not contribute significantly to the production. For compressible flow, is only
large in regions with high velocity divergence, such as at shocks.
The term in (Eqn. 25) is based on the “frozen stress” assumption [54]. This prevents the
values of and becoming too large through shocks, a situation that becomes
progressively worse as the mesh is refined at shocks. The parameter Compressible
Production (accessible on the Advanced Control part of the Turbulence section in ANSYS
CFX-Pre (For details, see Turbulence (p. 151 in "ANSYS CFX-Pre User's Guide").)) can be used
to set the value of the factor in front of , the default value is 3, as shown. A value of 1 will
provide the same treatment as CFX-4.
BuoyancyTurbulence
If the full buoyancy model is being used, the buoyancy production term is modeled as:
(Eqn. 26)
µt k ε
µt Cµρk2
ε-----=
Cµ
k ε
ρk( )∂t∂
-------------- ρUk( )∇•+ µµtσk-----+⎝ ⎠
⎛ ⎞ k∇∇• Pk ρε–+=
ρε( )∂t∂
-------------- ρUε( )∇•+ µµtσε-----+⎝ ⎠
⎛ ⎞ ε∇∇• εk--- Cε1Pk Cε2ρε–( )+=
Cε1 Cε2 σk σε
Pk
Pk µt U U∇ U∇ T+( )• 2
3-- U 3µt U∇• ρk+( )∇•–∇ Pkb+=
U∇•
U∇•
3µt
k ε
µt
Pkb
Pkbµt
ρσρ---------– g ρ∇•=
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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and if the Boussinesq buoyancy model is being used, it is:
(Eqn. 27)
This buoyancy production term is included in the equation if the Buoyancy Turbulence
option in ANSYS CFX-Pre is set to Production. It is also included in the equation if the
option is set to Production and Dissipation and if is positive:
(Eqn. 28)
If the directional option is enabled, then is modified by a factor accounting for the angle
between velocity and gravity vectors:
(Eqn. 29)
Default model constants are given by:
= 0.9 for Boussinesq buoyancy, = 1 for full buoyancy model
C3 = 1
Directional Dissipation = Off
For omega based turbulence models, the buoyancy turbulence terms for the equation
are derived from and according to the transformation .
The RNGk-epsilon Modelin ANSYS CFX
The RNG model is based on renormalization group analysis of the Navier-Stokes
equations. The transport equations for turbulence generation and dissipation are the same
as those for the standard - model, but the model constants differ, and the constant
is replaced by the function .
The transport equation for turbulence dissipation becomes:
(Eqn. 30)
where:
(Eqn. 31)
Pkbµt
ρσρ---------ρβg T∇•=
k
ε
Pkb
Pεb C3 max 0 Pkb,( )⋅=
Pεb
α
Pεb C3 max 0 Pkb,( ) αsin⋅⋅=
σρ σρ
ω
Pkb Pεb ε β'ωk=
k-ω
k ε Cε1
Cε1RNG
ρε( )∂t∂
-------------- ρUε( )∇•+ µµt
σεRNG---------------+⎝ ⎠
⎛ ⎞ ε∇∇• εk--- Cε1RNGPk Cε2RNGρε–( )+=
Cε1RNG 1.42 f η–=
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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and:
(Eqn. 32)
For details, see List of Symbols (p. 2).
The k-omegaModel in ANSYSCFX
One of the advantages of the - formulation is the near wall treatment for low-Reynolds
number computations. The model does not involve the complex non-linear damping
functions required for the - model and is therefore more accurate and more robust. A
low-Reynolds - model would typically require a near wall resolution of , while a
low-Reynolds number - model would require at least . In industrial flows, even
cannot be guaranteed in most applications and for this reason, a new near wall
treatment was developed for the - models. It allows for smooth shift from a
low-Reynolds number form to a wall function formulation.
The - models assumes that the turbulence viscosity is linked to the turbulence kinetic
energy and turbulent frequency via the relation:
(Eqn. 33)
The Wilcox k-omega Model
The starting point of the present formulation is the - model developed by Wilcox [11]. It
solves two transport equations, one for the turbulent kinetic energy, , and one for the
turbulent frequency, . The stress tensor is computed from the eddy-viscosity concept.
-equation:
(Eqn. 34)
-equation:
(Eqn. 35)
f η
η 1 η4.38----------–⎝ ⎠
⎛ ⎞
1 βRNGη3+( )
---------------------------------
ηPk
ρCµRNGε-----------------------=
=
k ω
k ε
k ε y+ 0.2<
k ω y+ 2<
y+ 2<
k ω
k ω
µt ρ kω----=
k ω
k
ω
k
ρk( )∂t∂
-------------- ρUk( )∇•+ µµtσk-----+⎝ ⎠
⎛ ⎞ ∇k∇• Pk β′ρkω–+=
ω
ρω( )∂t∂
--------------- ρUω( )∇•+ µµtσω------+⎝ ⎠
⎛ ⎞ ∇ω∇• αωk----Pk βρω2
–+=
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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In addition to the independent variables, the density, , and the velocity vector, , are
treated as known quantities from the Navier-Stokes method. is the production rate of
turbulence, which is calculated as in the - model (Eqn. 25).
The model constants are given by:
(Eqn. 36)
(Eqn. 37)
(Eqn. 38)
(Eqn. 39)
(Eqn. 40)
The unknown Reynolds stress tensor, , is calculated from:
(Eqn. 41)
In order to avoid the build-up of turbulent kinetic energy in stagnation regions, two
production limiters are available. For details, see Production Limiters (p. 82).
If is positive, the buoyancy production term is included in the equation if the
Buoyancy Turbulence option in ANSYS CFX-Pre is set to Production. It is also included in the
equation if the Option is set to Production and Dissipation. For details, see Buoyancy
Turbulence (p. 76).
The Baseline(BSL) k-OmegaModel
The main problem with the Wilcox model is its well known strong sensitivity to freestream
conditions (Menter [12]). Depending on the value specified for at the inlet, a significant
variation in the results of the model can be obtained. This is undesirable and in order to
solve the problem, a blending between the - model near the surface and the - model
in the outer region was developed by Menter [9]. It consists of a transformation of the -
model to a - formulation and a subsequent addition of the corresponding equations.
The Wilcox model is thereby multiplied by a blending function and the transformed -
model by a function . is equal to one near the surface and switches over to zero
inside the boundary layer (i.e., a function of the wall distance. For details, see Wall Distance
Formulation (p. 117). At the boundary layer edge and outside the boundary layer, the
standard - model is therefore recovered.
ρ UPk
k ε
β′ 0.09=
α 5 9⁄=
β 0.075=
σk 2=
σω 2=
τ
τ µt2s ρ23--δk–=
Pkb k
ω
ω
k ω k ε
k ε
k ω
F1 k ε
1 F1– F1
k ε
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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Wilcox model:
(Eqn. 42)
(Eqn. 43)
Transformed model:
(Eqn. 44)
(Eqn. 45)
Now the equations of the Wilcox model are multiplied by function , the transformed -
equations by a function and the corresponding - and - equations are added to
give the BSL model:
(Eqn. 46)
(Eqn. 47)
The coefficients of the new model are a linear combination of the corresponding
coefficients of the underlying models:
(Eqn. 48)
ρk( )∂t∂
-------------- ρUk( )∇•+ µµt
σk1--------+⎝ ⎠
⎛ ⎞ ∇k∇• Pk β′ρkω–+=
ρω( )∂t∂
--------------- ρUω( )∇•+ µµt
σω1---------+⎝ ⎠
⎛ ⎞ ∇ω∇• α1ωk----Pk β1ρω2
–+=
k-ε
ρk( )∂t∂
-------------- ρUk( )∇•+ µµt
σk2--------+⎝ ⎠
⎛ ⎞ ∇k∇• Pk β′ρkω–+=
ρω( )∂t∂
--------------- ρUω( )∇•+
µµt
σω2---------+⎝ ⎠
⎛ ⎞ ∇ω∇• 2ρ 1σω2 ω--------------- k ω∇∇ α2
ωk----Pk β2ρω2
–+ +⎝ ⎠⎛ ⎞=
F1 k ε
1 F1– k ω
ρk( )∂t∂
-------------- ρUk( )∇•+ µµt
σk3--------+⎝ ⎠
⎛ ⎞ ∇k∇• Pk β′ρkω–+=
ρω( )∂t∂
--------------- ρUω( )∇•+
µµt
σω3---------+⎝ ⎠
⎛ ⎞ ∇ω∇• 1 F1–( )2ρ 1σω2 ω--------------- k ω∇∇ α3
ωk----Pk β3ρω2
–+ +
Φ3 F1Φ1 1 F1–( )Φ2+=
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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All coefficients are listed again for completeness:
(Eqn. 49)
(Eqn. 50)
(Eqn. 51)
(Eqn. 52)
(Eqn. 53)
(Eqn. 54)
(Eqn. 55)
(Eqn. 56)
(Eqn. 57)
The Shear StressTransport (SST)
The - based SST model accounts for the transport of the turbulent shear stress and gives
highly accurate predictions of the onset and the amount of flow separation under adverse
pressure gradients.
The BSL model combines the advantages of the Wilcox and the - model, but still fails to
properly predict the onset and amount of flow separation from smooth surfaces. The
reasons for this deficiency are given in detail in Menter [9]. The main reason is that both
models do not account for the transport of the turbulent shear stress. This results in an
overprediction of the eddy-viscosity. The proper transport behavior can be obtained by a
limiter to the formulation of the eddy-viscosity:
(Eqn. 58)
where
(Eqn. 59)
Again is a blending function similar to , which restricts the limiter to the wall
boundary layer, as the underlying assumptions are not correct for free shear flows. is an
invariant measure of the strain rate.
β′ 0.09=
α1 5 9⁄=
β1 0.075=
σk1 2=
σω1 2=
α2 0.44=
β2 0.0828=
σk2 1=
σω2 1 0.856⁄=
k ω
k ε
vta1k
max a1ω SF2,( )--------------------------------------=
vt µt ρ⁄=
F2 F1
S
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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Blending FunctionsThe blending functions are critical to the success of the method. Their formulation is based
on the distance to the nearest surface and on the flow variables.
(Eqn. 60)
with:
(Eqn. 61)
where is the distance to the nearest wall, is the kinematic viscosity and:
(Eqn. 62)
(Eqn. 63)
with:
(Eqn. 64)
The Wall Scale EquationDuring the solution of a simulation using the SST or BSL model, you will see a plot in the
ANSYS CFX-Solver Manager for Wall Scale. These models require the distance of a node to
the nearest wall for performing the blending between - and - . Detailed information
on the wall scale equation is available. For details, see Wall Distance Formulation (p. 117).
ProductionLimiters
A disadvantage of standard two-equation turbulence models is the excessive generation of
turbulence energy, , in the vicinity of stagnation points. In order to avoid the build-up of
turbulent kinetic energy in stagnation regions, a formulation of limiters for the production
term in the turbulence equations is available.
The formulation follows Menter [9] and reads:
(Eqn. 65)
The coefficient is called Clip Factor and has a value of 10 for based models. This
limiter does not affect the shear layer performance of the model, but has consistently
avoided the stagnation point build-up in aerodynamic simulations.
F1 ar g14( )tanh=
ar g1 min max kβ′ωy------------ 500ν
y2ω------------,
⎝ ⎠⎜ ⎟⎛ ⎞ 4ρ k
CDkwσω2 y2------------------------------,
⎝ ⎠⎜ ⎟⎛ ⎞
=
y ν
CDkω max 2ρ 1σω2ω------------- k ω 1.0 10 10–×,∇∇⎝ ⎠
⎛ ⎞=
F2 ar g22( )tanh=
ar g2 max 2 kβ′ωy------------ 500v
y2ω-----------,
⎝ ⎠⎜ ⎟⎛ ⎞
=
k ε k ω
Pk
Pk min Pk Climρε,( )=
Clim ω
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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Using the standard Boussinesq-approximation for the Reynolds stress tensor, the
production term can be expressed for incompressible flow as:
(Eqn. 66)
where denotes the absolute value of the strain rate and the strain rate tensor.
Kato and Launder [128] noticed that the very high levels of the shear strain rate in
stagnation regions are responsible for the excessive levels of the turbulence kinetic energy.
Since the deformation near a stagnation point is nearly irrotational, i.e. the vorticity rate
is very small, they proposed the following replacement of the production term:
(Eqn. 67)
where denotes the absolute value of the vorticity rate and the vorticity tensor. In a
simple shear flow, and are equal. Therefore, formulation recovers in such flows, as seen
in the first parts of Equation 66 and Equation 67.
The production limiters described above are available for the two ε based turbulence
models and for the (k, )-, BSL- and SST-turbulence models. They are available in the
Advanced Control settings of the Turbulence Model section in ANSYS CFX-Pre. For details,
see Advanced Control (p. 152 in "ANSYS CFX-Pre User's Guide"). The allowed options of the
production limiter are Clip Factor and Kato Launder.
The Eddy Viscosity Transport Model
A very simple one-equation model has been developed by Menter [32] [33]. It is derived
directly from the model and is therefore named the model:
(Eqn. 68)
where is the kinematic eddy viscosity, is the turbulent kinematic eddy viscosity and
is a model constant.
Pk
Pk µtS2
=
S 2SijSij= Sij, 12--
U i∂xj∂
---------U j∂xi∂
--------+⎝ ⎠⎛ ⎞=
S Sij
S
Ω
Pk µtSΩ=
Ω 2ΩijΩij= Ωij, 12--
U i∂xj∂
---------U j∂xi∂
--------–⎝ ⎠⎛ ⎞=
Ω Ωij
S Ω
ω
k-ε k-ε( )1E
∂ρvt∂t
-----------∂ρU jvt
∂x j------------------+ c1ρvt S c2ρ
vtLvK--------⎝ ⎠
⎛ ⎞ 2µ
ρvtσ
--------+⎝ ⎠⎛ ⎞ ∂v
∂x j--------+–=
v vt σ
Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models
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The model contains a destruction term, which accounts for the structure of turbulence and
is based on the von Karman length scale:
(Eqn. 69)
where S is the shear strain rate tensor. The eddy viscosity is computed from:
(Eqn. 70)
In order to prevent the a singularity of the formulation as the von Karman length scale goes
to zero, the destruction term is reformulated as follows:
(Eqn. 71)
(Eqn. 72)
The coefficients are:
By default, the model is solved in combination with the scalable wall function. For details,
see Scalable Wall Functions (p. 108).
Coefficient Value
c1 0.144
c2 1.86
c3 7.0
A+ 13.5
0.41
1.0
LvK( )2 S2
∂S∂x j-------- ∂S
∂x j--------
-----------------=
µt ρvt=
Ek ε–
vtLvK--------⎝ ⎠
⎛ ⎞ 2=
EBB∂vt∂x j--------
∂vt∂x j--------=
E1e c3EBBEk ε–
c3EBB--------------⎝ ⎠
⎛ ⎞tanh=
∂ρvt∂t
-----------∂ρU jvt
∂x j------------------+ c1D1ρvtS c2ρE1e µ
ρvtσ
--------+⎝ ⎠⎛ ⎞ ∂vt
∂x j--------+–=
κ
σ
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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Low ReynoldsNumberFormulation
Low Reynolds formulation of the model is obtained by including damping functions. Near
wall damping functions have been developed to allow integration to the surface:
(Eqn. 73)
where D2 is required to compute the eddy-viscosity which goes into the momentum
equations:
(Eqn. 74)
The low Reynolds formulation of the model requires a near wall mesh resolution of
.
Reynolds Stress Turbulence Models
These models are based on transport equations for all components of the Reynolds stress
tensor and the dissipation rate. These models do not use the eddy viscosity hypothesis, but
solve an equation for the transport of Reynolds stresses in the fluid. The Reynolds stress
model transport equations are solved for the individual stress components.
Algebraic Reynolds stress models solve algebraic equations for the Reynolds stresses,
whereas differential Reynolds stress models solve differential transport equations
individually for each Reynolds stress component.
The exact production term and the inherent modeling of stress anisotropies theoretically
make Reynolds Stress models more suited to complex flows; however, practice shows that
they are often not superior to two-equation models.
The Reynolds averaged momentum equations for the mean velocity are:
(Eqn. 75)
where is a modified pressure, B is the sum of body forces and the fluctuating Reynolds
stress contribution is . Unlike eddy viscosity models, the modified pressure has no
turbulence contribution and is related to the static (thermodynamic) pressure by:
(Eqn. 76)
D1vt v+
vt v+-------------=
D2 1vt
A+κv-------------
⎝ ⎠⎜ ⎟⎛ ⎞ 2
–exp–=
µt ρD2vt=
k-ε( )1E
y+ 1<
ρU∂t∂
------------ ρU U⊗( ) µ U∇( )∇•–∇•+ p''∇– ρu u⊗( )∇•– B+=
p''
ρu u⊗
p'' p U∇• 23--µ ζ–⎝ ⎠
⎛ ⎞+=
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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In the differential stress model, is made to satisfy a transport equation. A separate
transport equation must be solved for each of the six Reynolds stress components of
. The differential equation Reynolds stress transport is:
(Eqn. 77)
where P and G are shear and buoyancy turbulence production terms of the Reynolds
stresses respectively, is the pressure-strain tensor, and C is a constant.
Buoyancy turbulence terms are controlled in the same way as for the and model.
See the text below (Eqn. 27).
The Reynolds Stress Model
The standard Reynolds Stress model in ANSYS CFX is based on the -equation. The ANSYS
CFX-Solver solves the following equations for the transport of the Reynolds stresses:
(Eqn. 78)
which can be written in index notation as:
(Eqn. 79)
where is the pressure-strain correlation, and P, the exact production term, is given by:
(Eqn. 80)
As the turbulence dissipation appears in the individual stress equations, an equation for
is still required. This now has the form:
(Eqn. 81)
u u⊗
ρu u⊗
ρu u⊗∂t∂
--------------------- ρu u⊗ U⊗( ) ρCkε--u u⊗ u u⊗∇( )
T
⎝ ⎠⎛ ⎞∇•–∇•+
P G φ 23--δρε–+ +⎝ ⎠
⎛ ⎞=
φ
k-ε k-ω
ε
ρu u⊗∂t∂
--------------------- ρU u u⊗⊗( )∇•+
P φ µ 23--csρ
k2
ε-----+⎝ ⎠
⎛ ⎞ u u⊗∇⎝ ⎠⎛ ⎞∇• 2
3--δρε–+ +⎝ ⎠
⎛ ⎞=
t∂∂ ρuiu j( )
xk∂∂ U kρuiu j( )+ Pij φij xk∂
∂ µ 23--csρ
k2
ε-----+⎝ ⎠
⎛ ⎞ uiu j∂xk∂
------------- 23--δijρε–+ +=
φij
P ρ u u⊗ U∇( )T U∇( )u u⊗+( )–=
ε
ρε( )∂t∂
--------------xk∂∂ ρU kε( )+
εk-- cε1P cε2ρε–( )
xk∂∂ µ
µtσε-----+⎝ ⎠
⎛ ⎞xk∂
∂ε+=
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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In these equations, the anisotropic diffusion coefficients of the original models are replaced
by an isotropic formulation, which increases the robustness of the Reynolds stress model.
The Reynolds Stress model is also available with anisotropic diffusion coefficients. In this
case, the ANSYS CFX-Solver solves the following equations for the transport of the Reynolds
stresses:
(Eqn. 82)
where is the pressure-strain correlation, and P, the exact production term, is given by
(Eqn. 80).
The equation for is:
(Eqn. 83)
The model constants are listed below for each model.
Pressure-StrainTerms
One of the most important terms in Reynolds stress models is the pressure-strain
correlation, .
The pressure strain correlations can be expressed in the general form:
(Eqn. 84)
where:
(Eqn. 85)
t∂∂ ρuiu j( )
xk∂∂ U kρuiu j( )+
Pij φij xk∂∂ µδkl csρ
kε--ukul+⎝ ⎠
⎛ ⎞ uiu j∂xl∂
------------- 23--δijρε–+ +
⎝ ⎠⎜ ⎟⎛ ⎞
=
φij
ε
ρε( )∂t∂
--------------xk∂∂ ρU kε( )+
εk-- cε1P cε2ρε–( )
xk∂∂ µδkl cερk
ε--ukul+⎝ ⎠
⎛ ⎞xl∂
∂ε+=
φij
φij φij1 φij2+=
φij1 ρε Cs1a Cs2 aa 13--a aδ•–⎝ ⎠
⎛ ⎞+⎝ ⎠⎛ ⎞–=
φij2 Cr1Pa– Cr2ρkS Cr3ρkS a a•–+=
Cr4ρk aST SaT 23--a–+ S• δ⎝ ⎠
⎛ ⎞ Cr5ρk aWT WaT+( )+ +
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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and:
(Eqn. 86)
In this formulation, a is the anisotropy tensor, S is the strain rate and W is the vorticity. This
general form can be used to model linear and quadratic correlations by using appropriate
values for the constants. The model constants are listed below.
AlternativeReynolds StressConstants
There are three varieties of the standard Reynolds stress models available. These are known
as LRR-IP, LRR-QI and SSG. Each model has different model constants.
The LRR-IP and LRR-QI models were developed by Launder, Reece and Rodi [4]. “IP” stands
for Isotropization of Production, and “QI” stands for Quasi-Isotropic. In these models, the
pressure-strain correlation is linear.
The SSG model was developed by Speziale, Sarkar and Gatski [5]. This model uses a
quadratic relation for the pressure-strain correlation.
The table below shows the values of the constants for each model.
Selection of the appropriate model is carried out on the Fluid Models panel of the Domains
form in ANSYS CFX-Pre. The following options correspond to the types of models listed
above:
• Reynolds Stress Model - LRR-IP
• SSG Reynolds Stress Model - SSG
• QI Reynolds Stress Model - LRR-IQ
a u u⊗k
---------------23--δ–=
S 12-- U∇ U∇( )T
+( )=
W 12-- U∇ U∇( )T
–( )=
Model C RSseRS cs c ce2
LRR-IP 0.1152 1.10 0.22 1.45 1.9
LRR-QI 0.1152 1.10 0.22 1.45 1.9
SSG 0.1 1.36 0.22 1.45 1.83
Model Cs1 Cs2 Cr1 Cr2 Cr3 Cr4 Cr5
LRR-IP 1.8 0.0 0.0 0.8 0.0 0.6 0.6
LRR-QI 1.8 0.0 0.0 0.8 0.0 0.873 0.655
SSG 1.7 -1.05 0.9 0.8 0.65 0.625 0.2
µ ε1
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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Omega-Based Reynolds Stress Models
ANSYS CFX provides two Reynolds Stress- models: the Omega Reynolds Stress and
Baseline (BSL) Reynolds Stress models. The two models relate to each other in the same way
as the two equation and BSL models. For details, see The Baseline (BSL) k-Omega
Model (p. 79).
The Reynolds Stress- turbulence model, or SMC- model, is a Reynolds Stress model
based on the -equation. The advantage of the -equation is that it allows for a more
accurate near wall treatment with an automatic switch from a wall function to a
low-Reynolds number formulation based on the grid spacing.
The modeled equations for the Reynolds stresses can be written as follows:
(Eqn. 87)
The OmegaReynolds StressModel
The Omega Reynolds Stress Model uses the following equation for :
(Eqn. 88)
The following coefficients apply:
(Eqn. 89)
The BSLReynolds StressModel
The coefficients and of the -equation, as well as both the turbulent Prandtl numbers
and , are blended between values from the two sets of constants, corresponding to
the -based model constants and the -based model constants transformed to an
-formulation:
(Eqn. 90)
ω
k-ω
ω ω
ω ω
∂ ρτij( )∂t
-----------------∂ U kρτij( )
∂xk------------------------+ ρPij–
23--β′ρωkδij ρΠij–
∂∂xk-------- µ
µtσ*------+⎝ ⎠
⎛ ⎞ ∂τij∂xk--------⎝ ⎠
⎛ ⎞+ +=
ω
∂ ρω( )∂t
---------------∂ U kρω( )
∂xk-----------------------+ αρω
k----Pk βρω2
–∂
∂xk-------- µ
µtσ-----+⎝ ⎠
⎛ ⎞ ∂ω∂xk--------⎝ ⎠
⎛ ⎞+=
σ∗ 2=
σ 2=
β 0.075=
α ββ′----- κ2
σ β′( )0.5-------------------– 5
9--= =
α β ω
σ* σ
ω ε
ω
ρω( )∂t∂
--------------- ∂ U kρω( )+
α3ωk----Pk β3ρω2
– ∂∂xk--------+ µ
µtσω3---------+⎝ ⎠
⎛ ⎞ ∂ω∂xk-------- 1 F1–( )2ρ 1
σ2ω---------- ∂k
∂xk-------- ∂ω
∂xk--------+⎝ ⎠
⎛ ⎞=
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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• Set 1 (SMC- zone):
(Eqn. 91)
The value of here corresponds to the model. For details, see The Wilcox k-omega
Model (p. 78). The von Karman constant has a commonly used value of 0.41.
• Set 2 (SMC- zone):
(Eqn. 92)
The blending of coefficients is done by smooth linear interpolation with the same weight
function F as the one used in a cross-diffusion term of the -equation (Eqn. 90):
(Eqn. 93)
where with:
(Eqn. 94)
and
(Eqn. 95)
Pressure-straincorrelation
The constitutive relation for the pressure-strain correlation is given by:
(Eqn. 96)
ω
σ∗1 2=
σ1 2=
β1 0.075=
α1ββ′----- κ2
σ β′( )0.5-------------------– 0.553= =
β k-ωκ
ε
σ∗2 1.0=
σ2 0.856=
β2 0.0828=
α2ββ′----- κ2
σ β′( )0.5-------------------– 0.44= =
ω
φ3 F φ1⋅ 1 F–( )φ2+=
F arg4( )tanh=
arg min max kβ′ωy------------ , 500υ
y2ω------------
⎝ ⎠⎜ ⎟⎛ ⎞
, 4ρ k
CDkωσk ε– y2-----------------------------------
⎩ ⎭⎨ ⎬⎧ ⎫
=
CDkω max 2ρ 1σk ε– ω------------------ ∂k
∂x j-------- ∂ω
∂x j-------- , 10 10–
⎝ ⎠⎛ ⎞=
Πij
β′C1ω τij23--kδij+⎝ ⎠
⎛ ⎞ α Pij23--Pδij–⎝ ⎠
⎛ ⎞– β Dij23--Pδij–⎝ ⎠
⎛ ⎞– γ k Sij13--Skkδij–⎝ ⎠
⎛ ⎞–⎝ ⎠⎛ ⎞=
Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models
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The production tensor of Reynolds stresses is given by:
(Eqn. 97)
The tensor Dij, participating in the pressure-strain model (Eqn. 96), differs from the
production tensor in the dot-product indices:
(Eqn. 98)
The turbulent viscosity in the diffusion terms of the balance equations (Eqn. 87) and (Eqn.
88) is calculated in the same way as in the Wilcox model. For details, see The Wilcox
k-omega Model (p. 78).
(Eqn. 99)
The coefficients for the model are:
Wall BoundaryCondition
The SMC- model is used in combination with the automatic wall treatment developed for
the based models ( , BSL and SST). The formulation has been recalibrated to
ensure a constant wall shear stress and heat transfer coefficient under variable near wall
resolution.
Rotating Frame of Reference for Reynolds Stress Models
One of the advantages of Reynolds stress transport models, when compared to and
models, is their ability to simulate the additional anisotropy of the Reynolds stresses
due to the Coriolis forces appearing in the rotating frame of reference.
The necessary additional source terms to account for the Coriolis forces have been
implemented into ANSYS CFX for use with any of the available Reynolds stress transport
models. These terms are described in a book by Wilcox [30], and in more detail in a paper by
Launder, Tselepidakis and Younis [31].
0.09
C1 1.8
C2 0.52
Pij τik∂U j∂xk--------- τ jk
∂U i∂xk--------- ;+= P 1
2--Pkk=
Dij τik∂U k∂x j---------- τ jk
∂U k∂xi----------+=
k-ω
µT ρ kω----=
β′α 8 C2+( ) 11⁄
β 8C2 2–( ) 11⁄
γ 60C2 4–( ) 55⁄
ω
k-ω k-ω
k-ε
k-ω
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If the flow equations are written in the frame of the coordinate system, which rotates
relative to the steady inertial frame with the vector angular velocity , then one new source
term Gij has to be added directly to the right hand side of the transport equation for ,
(Eqn. 87):
(Eqn. 100)
where is a Levi-Chivita factor, equal to 1 if its indices i,j,k form an even permutation of
1,2,3, equal to -1 for an odd permutation of 1,2,3, and equal to 0 if any two indices are
equal.
Besides the absolute velocity gradient tensor, participating in the production tensor (Eqn.
97) and in the model equation for the pressure-strain correlation (Eqn. 96), is written in the
rotating frame as a sum of the strain rate tensor Sij :
(Eqn. 101)
and the rotation tensor :
(Eqn. 102)
This representation of the velocity gradient results in an apparent additional source term Gij
(Eqn. 100), coming from the production term ( ). That is why in reference [31] the
Coriolis source term Gij differs from (Eqn. 100) by an additional factor of 2.
ANSYS CFX Transition Model Formulation
The following three transition models are available in ANSYS CFX, which are also discussed
briefly in the solver modeling chapter:
• Zero equation model (i.e., prescribed intermittency)
• One equation model, which solves only the intermittency equation using a userspecified value of the transition onset Reynolds number (where the transition onsetmomentum thickness Reynolds number is treated as a constant)
• Two equation model, where both the intermittency and transition onset Reynolds
number equations are solved.
The models can be used with the BSL, SST and SAS-SST (beta feature) turbulence models. In
the following the focus is on the two equation model. The empirical correlation used by this
model has been formulated to cover standard bypass transition as well as flows in low
free-stream turbulence environments (Langtry and Menter correlation). This model is the
Ω
τij
Gij ρ Ωk τ jmεikm τimε jkm+( )⋅ ⋅=
εijk
Sij 0.5∂U i∂x j---------
∂U j∂xi---------+⎝ ⎠
⎛ ⎞=
Ωij
Ωij 0.5∂U i∂x j---------
∂U j∂xi---------–⎝ ⎠
⎛ ⎞ εijk Ωk⋅+=
-pPij
Turbulence and Wall Function Theory: ANSYS CFX Transition Model Formulation
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recommend transition model for general-purpose applications and has been validated
together with the SST turbulence model [[101], [102], [103]]. It should be noted that a few
changes have been made to this model compared to CFX-5.7 [[101],[102]) in order to
improve the transition predictions. These include:
1. An improved transition onset correlation that results in improved predictions for bothnatural and bypass transition.
2. A modification to the separation induced transition modification that prevents it fromcausing early transition near the separation point.
3. Some adjustments of the model coefficients in order to better account for flow history
effects on the transition onset location.
It is recommended to use the new formulation [[103]], although the CFX-5.7 formulation can
be recovered by specifying the optional parameter “Transition Model Revision = 0” in the
CCL in the following way: FLUID MODELS: … TURBULENCE MODEL: Option = SST TRANSITIONAL TURBULENCE: Option = Gamma Theta Model Transition Model Revision = 0 TRANSITION ONSET CORRELATION: Option = Langtry Menter END END END … END
The transport equation for the intermittency, , reads:
(Eqn. 103)
The transition sources are defined as follows:
(Eqn. 104)
where is the strain rate magnitude. is an empirical correlation that controls the
length of the transition region. The destruction/relaminarization sources are defined as
follows:
(Eqn. 105)
γ
ργ( )∂t∂
--------------ρU jγ( )∂
x j∂--------------------- Pγ 1 Eγ 1– Pγ 2 Eγ 2–
x j∂∂ µ
µtσγ------+⎝ ⎠
⎛ ⎞ γ∂x j∂
-------+ +=+
Pγ 1 2FlengthρS γ Fonset[ ]cγ 3 Eγ 1; Pγ 1γ= =
S Flength
Pγ 2 2cγ 1( )ρΩγ Fturb Eγ 2; cγ 2Pγ 2γ= =
Turbulence and Wall Function Theory: ANSYS CFX Transition Model Formulation
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where is the vorticity magnitude. The transition onset is controlled by the following
functions:
(Eqn. 106)
(Eqn. 107)
(Eqn. 108)
is the critical Reynolds number where the intermittency first starts to increase in the
boundary layer. This occurs upstream of the transition Reynolds number, , and the
difference between the two must be obtained from an empirical correlation. Both the
and correlations are functions of .
The constants for the intermittency equation are:
(Eqn. 109)
The modification for separation-induced transition is:
(Eqn. 110)
Ω
Reνρ y2S
µ------------ RT; ρk
µω-------= =
Fonset1Reν
2.193 Reθc⋅----------------------------=
Fonset2 min max Fonset1 Fonset14,( ) 2.0,( )=
Fonset3 max 1RT2.5-------⎝ ⎠
⎛ ⎞3
0,–⎝ ⎠⎛ ⎞=
Fonset max Fonset2 Fonset3– 0,( )=
Fturb e
RT
4------⎝ ⎠
⎛ ⎞4
–
=
Reθc
Reθt
Flength Reθc Reθt
cy1 0.03=
cy2 50=
cy3 0.5=
σy 1.0=
γ sep min 2 maxReν
3.235Reθc------------------------⎝ ⎠
⎛ ⎞ 1 0,– Freattach 2,⋅⎝ ⎠⎛ ⎞ Fθt=
Freattach e
RT
20------⎝ ⎠
⎛ ⎞4
–
=
γ eff max γ γ sep,( )=
Turbulence and Wall Function Theory: ANSYS CFX Transition Model Formulation
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The model constants in (Eqn. 110) have been adjusted from those of Menter et al. [[101]] in
order to improve the predictions of separated flow transition. The main difference is that the
constant that controls the relation between and was changed from 2.193, its
value for a Blasius boundary layer, to 3.235, the value at a separation point where the shape
factor is 3.5 (see, for example Figure 2 in Menter et al. [[101]]). The boundary condition for
at a wall is zero normal flux while for an inlet is equal to 1.0.
The transport equation for the transition momentum thickness Reynolds number, ,
reads:
(Eqn. 111)
The source term is defined as follows:
(Eqn. 112)
(Eqn. 113)
(Eqn. 114)
(Eqn. 115)
The model constants for the equation are:
(Eqn. 116)
The boundary condition for at a wall is zero flux. The boundary condition for at
an inlet should be calculated from the empirical correlation based on the inlet turbulence
intensity.
The model contains three empirical correlations. is the transition onset as observed in
experiments. This has been modified from Menter et al. [101] in order to improve the
predictions for natural transition. It is used in (Eqn. 112). is the length of the
Reν Reθc
γ
γ
Reθt
ρReθt( )∂t∂
----------------------ρU jReθt( )∂
x j∂-----------------------------+ Pθt x j∂
∂ σθt µ µt+( )Reθt∂
x j∂-------------
⎝ ⎠⎜ ⎟⎛ ⎞
+=
Pθt cθtρt--- Reθt Reθt–( ) 1.0 Fθt–( ) t; 500µ
ρU 2------------= =
Fθt min max Fwake e
yδ---⎝ ⎠
⎛ ⎞ 4–
1.0 γ 1 50⁄–1.0 1 50⁄–-------------------------⎝ ⎠
⎛ ⎞ 2–,⋅
⎝ ⎠⎜ ⎟⎛ ⎞
1.0,⎝ ⎠⎜ ⎟⎛ ⎞
=
θBLReθtµ
ρU-------------- δBL;
152-----θBL δ; 50Ωy
U------------- δBL⋅= = =
Reωρω y2
µ------------- Fwake; e
Reω
1 5×10--------------⎝ ⎠
⎛ ⎞ 2–
= =
Reθt
cθt 0.03 σθt; 2.0= =
Reθt Reθt
Reθt
Flength
Turbulence and Wall Function Theory: ANSYS CFX Transition Model Formulation
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transition zone and goes into (Eqn. 104). is the point where the model is activated in
order to match both and ; it goes into (Eqn. 107). At present, these empirical
correlations are proprietary and are not given in this manual.
(Eqn. 117)
The first empirical correlation is a function of the local turbulence intensity, , and the
Thwaites pressure gradient coefficient defined as:
(Eqn. 118)
where is the acceleration in the streamwise direction.
The transition model interacts with the SST turbulence model, as follows:
(Eqn. 119)
(Eqn. 120)
(Eqn. 121)
where and are the original production and destruction terms for the SST model and
is the original SST blending function. Note that the production term in the
-equation is not modified. The rational behind the above model formulation is given in
detail in Menter et al. [101].
In order to capture the laminar and transitional boundary layers correctly, the grid must
have a of approximately one. If the is too large (i.e., > 5) then the transition onset
location moves upstream with increasing . It is recommended to use the High Resolution
discretization scheme (which is a bounded second-order upwind biased discretization) for
the mean flow, turbulence and transition equations.
Reθc
Reθt Flength
Reθt f Tu λ,( ) Flength; f Reθt( ) Reθc, f Reθt( )= = =
Tu
λθ
λθθ2
v-----⎝ ⎠
⎛ ⎞ Udsd
-------=
Ud sd⁄
t∂∂ ρk( )
x j∂∂ ρu jk( )+ Pk Dk–
x j∂∂ µ σkµt+( ) k∂
x j∂-------⎝ ⎠
⎛ ⎞+=
Pk γ eff Pk Dk; min max γ eff 0.1,( ) 1.0,( )Dk= =
Ryρy k
µ------------- F3; e
Ry
120--------⎝ ⎠
⎛ ⎞8
–
F1; max Florig F3,( )= = =
Pk Dk
F1orig
ω
y+ y+
y+
Turbulence and Wall Function Theory: Large Eddy Simulation Theory
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Large Eddy Simulation Theory
This section outlines the theoretical details of the LES model in ANSYS CFX. Additional
information on setting up an LES simulation is available, as well as modeling advice. For
details, see The Large Eddy Simulation Model (LES) (p. 111 in "ANSYS CFX-Solver Modeling
Guide").
The non filtered Navier-Stokes equations are:
(Eqn. 122)
Large Eddy Simulation (LES) is about filtering of the equations of movement and
decomposition of the flow variables into a large scale (resolved) and a small scale
(unresolved) parts. Any flow variable f can be written such as:
(Eqn. 123)
where , the large scale part, is defined through volume averaging as:
(Eqn. 124)
where is the filter function (called the hat filter or Gaussian filter).
After performing the volume averaging and neglecting density fluctuations, the filtered
Navier-Stokes equations become:
(Eqn. 125)
The non linear transport term in the filtered equation can be developed as:
(Eqn. 126)
In time, averaging the terms (2) and (3) vanish, but when using volume averaging, this is no
longer true.
∂ ρU i( )∂t
-----------------∂ ρU iU j( )
∂x j------------------------+ ∂p
∂xi-------– µ
∂2U i∂x j∂x j----------------+=
f f f '+=
f
f xi t,( ) G xi xi'–( ) f xi' t,( ) xi'dVol∫=
G xi xi'–( )
∂ ρU i( )∂t
-----------------∂ ρU iU j( )
∂x j------------------------+ ∂ p
∂xi-------– µ
∂2U i∂x j∂x j----------------+=
U iU j U i ui'+( ) U j u j'+( )=
U iU j U iu j' U jui' ui'u j'+ + +=
(1) (2) (3) (4)
Turbulence and Wall Function Theory: Large Eddy Simulation Theory
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Introducing the sub-grid scale (SGS) stresses, , as:
(Eqn. 127)
you can rewrite the filtered Navier-Stokes equations as:
(Eqn. 128)
(Eqn. 129)
with:
(Eqn. 130)
(Eqn. 131)
Smagorinsky Model
The Smagorinsky model [34] can be thought of as combining the Reynolds averaging
assumptions given by Lij + Cij = 0 with a mixing-length based eddy viscosity model for the
Reynolds SGS tensor. It is thereby assumed that the SGS stresses are proportional to the
modulus of the strain rate tensor, of the filtered large-scale flow:
(Eqn. 132)
To close the equation, a model for the SGS viscosity vSGS is needed. Based on dimensional
analysis, the SGS viscosity can be expressed as:
(Eqn. 133)
where l is the length scale of the unresolved motion (usually the grid size )
and qSGS is the velocity of the unresolved motion.
τij
τij uiu j U iU j–=
∂ ρU i( )∂t
-----------------∂ ρτij ρU iU j+( )
∂x j----------------------------------------+ ∂ p
∂xi-------– µ
∂2U i∂x j∂x j----------------+=
∂ ρU i( )∂t
-----------------∂ ρU iU j( )
∂x j------------------------+ ∂ p
∂xi-------– µ
∂2U i∂x j∂x j----------------
∂ ρτij( )∂x j
-----------------–+=
τij uiu j U iU j–=
U iU j U iu j' U jui' ui'u j' U iU j–+ + +=
Lij Cij Rij+ +=
Cij U iu j' U jui'+ Cross Terms= =
Sij
τij13--τkk– 2 vSGS Sij⋅ ⋅– v– SGS
∂U i∂x j---------
∂U j∂xi---------+
⎝ ⎠⎜ ⎟⎛ ⎞
⋅= =
vSGS l qSGS∝
∆ Vol( )1 3⁄=
Turbulence and Wall Function Theory: Large Eddy Simulation Theory
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In the Smagorinsky model, based on an analogy to the Prandtl mixing length model, the
velocity scale is related to the gradients of the filtered velocity:
(Eqn. 134)
This yields the Smagorinsky model [34] for the SGS viscosity:
(Eqn. 135)
with CS the Smagorinsky constant. The value of the Smagorinsky constant for isotropic
turbulence with inertial range spectrum:
(Eqn. 136)
is:
(Eqn. 137)
For practical calculations, the value of CS is changed depending on the type of flow and
mesh resolution. Its value is found to vary between a value of 0.065 (channel flows) and 0.25.
Often a value of 0.1 is used.
Wall Damping
Close to walls, the turbulent viscosity can be damped using a combination of a mixing
length minimum function, and a viscosity damping function :
(Eqn. 138)
with:
(Eqn. 139)
CS and are constants which you can set; their default values are 0.1 and 0.4 respectively.
By default, the damping function is 1.0. A Van Driest and a Piomelli like damping can be
specified by the user. For the Van Driest case, the damping function is:
(Eqn. 140)
qSGS ∆ S= where S 2SijSij( )1 2⁄=
vSGS CS∆( )2 S=
E k( ) Ckε2 3⁄ k 5 3⁄–=
CS1π--- 2
3Ck---------⎝ ⎠
⎛ ⎞ 3 4⁄0.18= =
f µ
µT ρ min lmix f µCS∆,( )2 2SijSij⋅=
lmix κ ywall⋅=
κ
f µ
f µ 1 y A⁄–( )exp–=
Turbulence and Wall Function Theory: Detached Eddy Simulation Theory
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with A = 25. For the Piomelli case it is:
(Eqn. 141)
with A = 25. The normalized wall distance:
(Eqn. 142)
is defined as a function of the calculated wall distance , kinematic viscosity , and local
velocity scale .
The Van Driest or Piomelli wall damping can be switched on when the LES turbulence model
is selected. The damping factor A is defaulted to 25.0.
Detached Eddy Simulation Theory
This section outlines the theoretical details of the DES model in ANSYS CFX. Details on
setting up a DES simulation and modeling advice are available. For details, see The
Detached Eddy Simulation Model (DES) (p. 116 in "ANSYS CFX-Solver Modeling Guide").
Experience has shown that the use of LES in boundary layer flows at high Re numbers is
prohibitively expensive [57] and therefore not useful for many industrial flow simulations.
On the other hand, turbulent structures can be resolved in massively separated regions,
where the large turbulent scales are of the same dimension as the geometrical structure
generating them (airfoil flaps, buildings, blunt trailing edges on turbine blades). DES is an
attempt to combine elements of RANS and LES formulations in order to arrive at a hybrid
formulation, where RANS is used inside attached and mildly separated boundary layers.
Additionally, LES is applied in massively separated regions. While this approach offers many
advantages, it is clearly not without problems, as the model has to identify the different
regions automatically. DES models require a detailed understanding of the method and the
grid generation requirements and should not be used as “black-box.” You are advised to
read the CFX technical report [55] on the subject, which explains all details on grid
generation requirements, zonal formulation and boundary conditions.
The ANSYS CFX version of the DES model is based on the SST formulation. The advantage of
this combination is that the accurate prediction of turbulent boundary layers up to
separation and in mildly separated regions carries over from the SST model. In addition, the
SST model supports the formulation of a zonal DES formulation [56], which is less sensitive
to grid resolution restrictions than the standard DES formulation, as proposed by Strelets
[58]. Refer to the CFX technical report [55] for details.
f µ 1 y A⁄–( )3[ ]exp–=
y y u⋅( ) v⁄=
y ν
u
Turbulence and Wall Function Theory: Detached Eddy Simulation Theory
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SST-DES Formulation Strelets et al.
The idea behind the DES model of Strelets [58] is to switch from the SST-RANS model to an
LES model in regions where the turbulent length, Lt, predicted by the RANS model is larger
than the local grid spacing. In this case, the length scale used in the computation of the
dissipation rate in the equation for the turbulent kinetic energy is replaced by the local grid
spacing, .
(Eqn. 143)
The practical reason for choosing the maximum edge length in the DES formulation is that
the model should return the RANS formulation in attached boundary layers. The maximum
edge length is therefore the safest estimate to ensure that demand.
The DES modification of Strelets can be formulated as a multiplier to the destruction term in
the k-equation:
(Eqn. 144)
with CDES equal to 0.61, as the limiter should only be active in the model region. The
numerical formulation is also switched between an upwind biased and a central difference
scheme in the RANS and DES regions respectively.
Zonal SST-DES Formulation in ANSYS CFX
The main practical problem with the DES formulation (both for the Spalart Allmaras and the
standard SST-DES model) is that there is no mechanism for preventing the limiter from
becoming active in the attached portion of the boundary layer. This will happen in regions
where the local surface grid spacing is less than the boundary layer thickness with
c of the order of one. In this case the flow can separate as a result of the grid spacing
(grid-induced separation), which is undesirable. In order to reduce this risk, ANSYS CFX
offers a zonal formulation of the DES formulation, based on the blending functions of the
SST model [56]. The blending functions are functions of the wall distance. For details, see
Wall Distance Formulation (p. 117).
(Eqn. 145)
∆
ε β∗kω k32--
Lt⁄ k32--
CDES∆( )⁄ for CDES∆ Lt<( )→= =
∆ max ∆i( ) ; Lt k( ) β⁄ ∗ω= =
ε β∗kω β∗kω FDES with FDES maxLt
CDES∆---------------- 1( , )=⋅→=
k ε–
∆s cδ<
FDES CFX– maxLt
CDES∆---------------- 1 FSST–( ) 1( , ) with FSST 0 F1 F2, ,==
Turbulence and Wall Function Theory: Detached Eddy Simulation Theory
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In case FSST is set to 0, the Strelets model is recovered. The default option is ,
which offers the highest level of protection against grid-induced separation, but might also
be less responsive in LES regions. For details, refer to the CFX technical report [55].
Discretization of the Advection Terms
Following Strelets [58], the zonal DES model switches from a second-order upwind scheme
in the RANS region to a central difference scheme in the LES region. The finite volume
method approximates the advection term in the transport equation for a generic variable
as:
(Eqn. 146)
where is the control volume value, the index denotes the integration points on the
control volume faces, and are the mass flow and the transported variable value
estimated at the integration point . The value of is obtained by interpolating from the
surrounding grid nodes according to the selected discretization scheme. Blending between
the upwind-biased scheme and the central scheme is achieved by combining the
corresponding interpolation values and using the blending function :
(Eqn. 147)
A specific form of the blending function is taken from Strelets [58] with only minor
changes:
(Eqn. 148)
(Eqn. 149)
(Eqn. 150)
FSST F2=
ϕ
∇ ρUϕ⋅ 1ΩCV----------- mipϕip
ip∑≈
ΩCV ip
mip ϕip
ip ϕip
ϕip U, ϕip C, σ
ϕip σ ϕip U, 1 σ–( ) ϕip C,⋅+⋅=
σ
σ σmax ACH1( )tanh⋅=
A CH2 maxCDES∆max
Lturb g------------------------ 0.5– 0,⎝ ⎠
⎛ ⎞⋅=
Lturbk
cµω---------=
g max B4( )tanh 10 10–,( )=
B CH3Ωmax Ω S,( )
max S2 Ω2+2
------------------ 10 10–,⎝ ⎠⎛ ⎞
---------------------------------------------------⋅=
Turbulence and Wall Function Theory: Scale-Adaptive Simulation Theory
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with being the maximum neighboring grid edge, and the same constant values as in
Strelets [58]:
(Eqn. 151)
Note: For the SST-DES model, the blending function is additionally limited in order toprevent the grid-induced separation:
(Eqn. 152)
where F1 and F2 are the first and second SST blending function and ([129]),
respectively. BF1 and BF2 are blend factors and are used in order to select one of the two
blending functions. The default and the recommendation is to use the second SST blending
function F2, therefore the corresponding default values are BF1 = 0 and BF2 = 1.
A limiter based on the Courant-number has been developed in order to avoid oscillations
due to the central difference scheme, which can occur for medium and high
Courant-numbers (this limiter is not given in the original formulation of Strelets [58]). The
blending function is limited in the following way:
(Eqn. 153)
The default values of and are 5.0 and 1.0, respectively.
Boundary Conditions
For LES simulations, unsteady fluctuations have to be specified in most cases at the inlet
boundaries. This greatly complicates the use of LES in industrial flows, as the details of these
fluctuations are generally not known. In DES simulations, it is in most cases possible to
specify the same boundary conditions as for the SST model. This is particularly true if the
default setting of the zonal model are used ([55]).
Scale-Adaptive Simulation Theory
The Scale-Adaptive Simulation (SAS) is an improved URANS formulation, which allows the
resolution of the turbulent spectrum in unstable flow conditions. The SAS concept is based
on the introduction of the von Karman length-scale into the turbulence scale equation. The
information provided by the von Karman length-scale allows SAS models to dynamically
adjust to resolved structures in a URANS simulation, which results in a LES-like behavior in
unsteady regions of the flowfield. At the same time, the model provides standard RANS
capabilities in stable flow regions.
∆max
σmax 1= CH1, 3 CH2, 1 CH3, 2= = =
σ
σSST-DES max σ BF1 F1 BF2 F2⋅,⋅,( )=
F1 F2
σ
σlim max σ 1 min CFLmax
CFL------------------ 1,⎝ ⎠
⎛ ⎞CFLEXP
–,=
CFLmax CFLEXP
Turbulence and Wall Function Theory: Scale-Adaptive Simulation Theory
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The first model with SAS capability was based on the one-equation model (see The
Eddy Viscosity Transport Model (p. 83)) for the eddy-viscosity ([132]). The model is
a direct result of the transformation of the high Reynolds number version of the model
to a one-equation model, using equilibrium assumptions. During the transformation, the
von Karman length-scale, , appears in the equation in the sink term. It turned out that
this new term introduces a dynamic behavior into the model, which was not present in the
underlying two-equation model.
This leads one to wonder if could also be derived naturally within the more general
two-equation framework. It was shown by Menter and Egorov [130] that the exact transport
equation for the turbulent length-scale, as derived by Rotta [134], does actually introduce
the second derivative of the velocity field and thereby into the turbulent scale
equation. In Menter and Egorov [130], a two-equation turbulence model was presented
using a formulation, which can be operated in RANS and SAS mode. While the further
development of this model is still ongoing, it was considered desirable to investigate if the
SAS term in the model could be transformed to existing two-equation models. The
target two-equation model is the SST model (see [129], [132], and The Eddy Viscosity
Transport Model (p. 83)), using the -equation as the second scale equation. The
transformation of the SAS term to the SST model will be presented in the next section
(following Menter and Egorov [131]). This model is called the SAS-SST model.
SAS-SST Model Formulation
The starting point of the transformation to the SST model is the formulation as given
by Menter and Egorov [130]. The following equations have been derived there for the
variables and :
(Eqn. 154)
with:
(Eqn. 155)
k-ε( )1E
k-ε( )1E
k-ε
LvK
LvK
LvK
k-νt
k-νt
ω
k-νt
k Φ kL=
ρk∂t∂
---------U jρk∂
x j∂----------------+ Pk cµ
3 4⁄ ρk2
Φ----- ∂
y∂------
µtσk------ k∂
y∂------+–=
ρΦ∂t∂
-----------U jρΦ∂
x j∂------------------+ ζ1
φk---Pk ζ2µtS U″ Φ2
k3 2⁄---------- ζ3ρk ∂y∂
------µtσΦ------- Φ∂
y∂-------+––=
vt cµ1 4⁄ Φ=
U″∂2U i
∂x j2
-----------∂2U i
∂x j2
-----------=
Turbulence and Wall Function Theory: Scale-Adaptive Simulation Theory
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where is the absolute value of the strain-rate, the production rate of the turbulent
kinetic energy, and .
The determination of the constants mainly followed Rotta's estimates resulting in
and . comes from the demand that the equations have to satisfy
the logarithmic law of the wall. The diffusion constants are chosen as to
ensure a proper behavior of the equations at a viscous-inviscid interface.
The SAS-relevant term in the equation for is the term with the second derivative . As
a result of this term, the length-scale, , predicted by the above model is largely
proportional to the von Karman length-scale:
(Eqn. 156)
It has been demonstrated by Menter and Egorov [130] that the term is the result of the
exact length-scale equation of Rotta [134]. It therefore has a strong theoretical foundation
and is one of the central terms in Equation 154. It has also been shown that this term allows
the model to operate in a scale-adaptive simulation (SAS) mode. The reason is that
adjusts to the already resolved scales in a simulation and provides a length-scale, which is
proportional to the size of the resolved eddies. Standard turbulence models, on the other
hand, always provide a length-scale proportional to the thickness of the shear layer. They do
not adjust to the local flow topology and are therefore overly diffusive.
In order to provide the SAS capability to the SST model, the -equation is transformed to
the framework using:
(Eqn. 157)
The resulting -equation reads:
(Eqn. 158)
S Pk
cµ 0.09= κ, 0.41=
ζ1 0.8=
ζ3 0.0326= ζ2 3.51=
σk σΦ 2 3⁄= =
Φ U″
L
LvK κ U∂ y∂⁄∂2U y∂ 2⁄----------------------=
U″
LvK
Φ
k-ω
Φ 1
cµ1 4⁄--------- k
ω----=
ω
∂ρωt∂
-----------∂U jρω
x j∂-----------------+ αρS2 βρω2 ∂
x j∂-------
µtσω------∂ω
x j∂-------⎝ ⎠
⎛ ⎞+–=
+ 2ρσΦ------- 1
ω---- k∂
x j∂-------∂ω
x j∂------- k
ω2------∂ω
x j∂-------∂ω
x j∂-------–⎝ ⎠
⎛ ⎞ ζ2˜ κρS2 L
LvK--------+
+ ρωk
------- ∂x j∂
-------vtσω------ k∂
x j∂-------⎝ ⎠
⎛ ⎞ 1σk------ 1
σΦ-------–⎝ ⎠
⎛ ⎞
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with and . The first three terms on the right hand side of
Equation 158 are the standard terms of the original Wilcox model [136]. The term
is the cross-diffusion term, which would also result from the transformation of
the model to the model. It is already included in a zonal way in the SST model and
helps the model to prevent the freestream sensitivity of the Wilcox model [129]. The last
term in parentheses is zero, as both the model and the model use identical
diffusion constants in both equations. The remaining terms are therefore:
(Eqn. 159)
It is the goal of the transformation to preserve the SST model in the RANS regime and to
activate the SAS capability in URANS regions. In RANS regime (and particularly in boundary
layers) the two terms on the right hand side of Equation 159 are of the same size, whereas
the term dominates in the SAS regime:
(Eqn. 160)
In order to preserve the SST model in the RANS region, the term is modeled as follows
(note that the term has little influence in the SAS regime, as it is also dominated
by the term):
(Eqn. 161)
The constants are taken from the model with minor adjustments:
(Eqn. 162)
The constant has been introduced for the calibration of the SAS term in the SST
environment. Finally, the term is added to the right hand side of the -equation
of the SST model.
ζ2˜ ζ2 cSAS⋅= L k cµ
0.25ω( )⁄=
2ρσΦ------- 1
ω---- k∂
x j∂-------∂ω
x j∂-------
k-ε k-ω
k-ω k-Φ
FSST-SAS2ρσΦ------- k
ω2------∂ω
x j∂-------∂ω
x j∂-------– ζ2
˜ κρS2 LLvK--------+=
LvK
2ρσΦ------- k
ω2------∂ω
x j∂-------∂ω
x j∂------- ζ2
˜ κρS2 LLvK--------≈ (RANS regime)
2ρσΦ------- k
ω2------∂ω
x j∂-------∂ω
x j∂------- ζ2
˜ κρS2 LLvK--------< (SAS regime)
FSST
1k2----- ∂k
x j∂------- ∂k
x j∂-------
ζ2˜
FSST-SAS =
ρ FSASmax ζ2˜ κS2 L
LvK-------- 2
σΦ-------k max 1
ω2------∂ω
x j∂-------∂ω
x j∂------- 1
k2----- ∂k
x j∂------- ∂k
x j∂-------,⎝ ⎠
⎛ ⎞⋅– 0,⋅
k-Φ
FSAS 1.25= ζ2˜ ζ2 cSAS⋅ 1.755= = σΦ, , 2 3⁄=
FSAS
FSST-SAS ω
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It should be noted that the details of the formulation of the second derivative term are
important. The term has to be formulated as given in Equation 155. Simpler formulations,
based on the strain rate are not representative of the smallest scales in the SAS regime; e.g.:
(Eqn. 163)
as can easily be verified by assuming a harmonic variation of the velocity field down to the
grid limit.
Similar to the DES formulation, the SAS model also benefits from a switch in the numerical
treatment between the steady and the unsteady regions. In DES, this is achieved by a
blending function as proposed by Strelets [58], which allows the use of a second order
upwind scheme with the ANSYS CFX-Solver in RANS regions and a second order central
scheme in unsteady regions. The blending functions are based on several parameters,
including the grid spacing and the ratio of vorticity and strain rate, as explained below.
Discretization ofthe AdvectionTerms
The discretization of the advection is the same as that for the SST-DES model, beside the fact
that no RANS-shielding is performed for the SAS-SST model. For details, see Discretization
of the Advection Terms (p. 102).
Modeling Flow Near the Wall
This section presents the mathematical details of how flow near to a no-slip wall is modeled
in ANSYS CFX. An introduction to near-wall flow, modeling details and guidelines on using
wall functions are presented. For details, see Modeling Flow Near the Wall (p. 125 in "ANSYS
CFX-Solver Modeling Guide").
Mathematical Formulation
The wall-function approach in ANSYS CFX is an extension of the method of Launder and
Spalding [13]. In the log-law region, the near wall tangential velocity is related to the
wall-shear-stress, , by means of a logarithmic relation.
In the wall-function approach, the viscosity affected sublayer region is bridged by
employing empirical formulas to provide near-wall boundary conditions for the mean flow
and turbulence transport equations. These formulas connect the wall conditions (e.g., the
wall-shear-stress) to the dependent variables at the near-wall mesh node which is
presumed to lie in the fully-turbulent region of the boundary layer.
The logarithmic relation for the near wall velocity is given by:
(Eqn. 164)
U″∂2U i
∂x j2
-----------∂2U i
∂x j2
-----------=∂S∂x j-------- ∂S
∂x j--------≠
τω
u+ U tuτ------
1κ--- y+( )ln C+= =
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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where:
(Eqn. 165)
(Eqn. 166)
u+ is the near wall velocity, is the friction velocity, Ut is the known velocity tangent to the
wall at a distance of from the wall, y+ is the dimensionless distance from the wall, is
the wall shear stress, is the von Karman constant and C is a log-layer constant depending
on wall roughness (natural logarithms are used).
A definition of in the different wall formulations is available. For details, see Solver Yplus
and Yplus (p. 109).
Scalable WallFunctions
(Eqn. 164) has the problem that it becomes singular at separation points where the near wall
velocity, Ut, approaches zero. In the logarithmic region, an alternative velocity scale, u* can
be used instead of u+:
(Eqn. 167)
This scale has the useful property that it does not go to zero if Ut goes to zero (in turbulent
flow k is never completely zero). Based on this definition, the following explicit equation for
can be obtained:
(Eqn. 168)
The absolute value of the wall shear stress , is then obtained from:
(Eqn. 169)
where:
(Eqn. 170)
and u* is as defined earlier.
y+ ρ∆yuτµ
----------------=
uττωρ-----⎝ ⎠
⎛ ⎞1 2⁄
=
uτ
∆y τω
κ
∆y
u* Cµ1 4⁄ k1 2⁄
=
uτ
uτU t
1κ--- y*( )ln C+------------------------------=
τω
τω ρu*uτ=
y* ρu*∆y( ) µ⁄=
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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One of the major drawbacks of the wall-function approach is that the predictions depend
on the location of the point nearest to the wall and are sensitive to the near-wall meshing;
refining the mesh does not necessarily give a unique solution of increasing accuracy
(Grotjans and Menter [10]). The problem of inconsistencies in the wall-function, in the case
of fine meshes, can be overcome with the use of the Scalable Wall Function formulation
developed by ANSYS CFX. It can be applied on arbitrarily fine meshes and allows you to
perform a consistent mesh refinement independent of the Reynolds number of the
application.
The basic idea behind the scalable wall-function approach is to limit the y* value used in the
logarithmic formulation by a lower value of . 11.06 is the intersection
between the logarithmic and the linear near wall profile. The computed is therefore not
allowed to fall below this limit. Therefore, all mesh points are outside the viscous sublayer
and all fine mesh inconsistencies are avoided.
It is important to note the following points:
• To fully resolve the boundary layer, you should put at least 10 nodes into the boundarylayer.
• Do not use Standard Wall Functions unless required for backwards compatibility.
• The upper limit for y+ is a function of the device Reynolds number. For example, a large
ship may have a Reynolds number of 109 and y+ can safely go to values much greater
than 1000. For lower Reynolds numbers (e.g., a small pump), the entire boundary layer
might only extend to around y+ = 300. In this case, a fine near wall spacing is required to
ensure a sufficient number of nodes in the boundary layer.
If the results deviate greatly from these ranges, the mesh at the designated Wall boundaries
will require modification, unless wall shear stress and heat transfer are not important in the
simulation.
Solver Yplusand Yplus
In the solver output, there are two arrays for the near wall spacing. The definition for the
Yplus variable that appears in the post processor is given by the standard definition of
generally used in CFD:
(Eqn. 171)
where is the distance between the first and second grid points off the wall.
In addition, a second variable, Solver Yplus, is available which contains the used in the
logarithmic profile by the solver. It depends on the type of wall treatment used, which can
be one of three different treatments in ANSYS CFX. They are based on different distance
definitions and velocity scales. This has partly historic reasons, but is mainly motivated by
the desire to achieve an optimum performance in terms of accuracy and robustness:
• Standard wall function (based on )
y* max y* 11.06( , )=
y*
y+
y+
y+ τω ρ⁄ ∆n⋅v
------------------------------=
∆n
y+
∆y ∆n/4=
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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• Scalable wall function (based on )
• Automatic wall treatment (based on )
The scalable wall function y+ is defined as:
(Eqn. 172)
and is therefore based on 1/4 of the near wall grid spacing.
Note that both the scalable wall function and the automatic wall treatment can be run on
arbitrarily fine meshes.
AutomaticNear-WallTreatment forOmega-BasedModels
While the wall-functions presented above allow for a consistent mesh refinement, they are
based on physical assumptions which are problematic, especially in flows at lower Reynolds
numbers (Re<105), as the sublayer portion of the boundary layer is neglected in the mass
and momentum balance. For flows at low Reynolds numbers, this can cause an error in the
displacement thickness of up to 25%. It is therefore desirable to offer a formulation which
will automatically switch from wall-functions to a low-Re near wall formulation as the mesh
is refined. The model of Wilcox has the advantage that an analytical expression is
known for in the viscous sublayer, which can be exploited to achieve this goal. The main
idea behind the present formulation is to blend the wall value for between the
logarithmic and the near wall formulation. The flux for the k-equation is artificially kept to
be zero and the flux in the momentum equation is computed from the velocity profile. The
equations are as follows:
Flux for the momentum equation, FU:
(Eqn. 173)
with:
(Eqn. 174)
(Eqn. 175)
Flux for the k-equation:
(Eqn. 176)
∆y ∆n/4=
∆y ∆n=
y+ max y* 11.06( , )= y* u*∆n 4⁄v
--------------------=
k-ω
ω
ω
FU ρuτ u*–=
uτ v ∆U∆y--------=
u* max a1k uτ( , )=
Fk 0=
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In the -equation, an algebraic expression is specified instead of an added flux. It is a blend
between the analytical expression for in the logarithmic region:
(Eqn. 177)
and the corresponding expression in the sublayer:
(Eqn. 178)
with being the distance between the first and the second mesh point. In order to
achieve a smooth blending and to avoid cyclic convergence behavior, the following
formulation is selected:
(Eqn. 179)
While in the wall-function formulation, the first point is treated as being outside the edge of
the viscous sublayer, the location of the first mesh point is now virtually moved down
through the viscous sublayer as the mesh is refined in the low-Re mode. It is to be
emphasized, that the physical location of the first mesh point is always at the wall (y = 0). The
error in the wall-function formulation results from this virtual shift, which amounts to a
reduction in displacement thickness. This error is always present in the wall-function mode,
but is reduced to zero as the method shifts to the low-Re model. The shift is based on the
distance between the first and the second mesh point y = y2 - y1 with y being the wall
normal distance.
Treatment ofRough Walls
The above wall function equations are appropriate when the walls can be considered as
hydraulically smooth. For rough walls, the logarithmic profile still exists, but moves closer to
the wall. Roughness effects are accounted for by modifying the expression for u+ as follows:
(Eqn. 180)
where:
(Eqn. 181)
and yR is the equivalent sand grain roughness [14].
ω
ω
ωlu*
a1κy------------
1a1κv------------ u*2
y+--------= =
ωs6v
β ∆y( )2-----------------=
∆y
ωω ωs 1ωlωs-----⎝ ⎠
⎛ ⎞ 2+=
∆
u+ 1κ---
y*1 0.3k++----------------------⎝ ⎠
⎛ ⎞ln C+=
k+ yRρµ---u*=
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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You are advised to exercise care in the use of the rough wall option together with the
standard wall-function approach. Inaccuracies can arise if an equivalent sand grain
roughness is of the same order, or larger than the distance from the wall to the first node.
The first element off the wall should not be much thinner than the equivalent sand grain
roughness. If this element becomes too thin, then negative values of u+ can be calculated,
resulting in the ANSYS CFX-Solver failing. For details, see Wall Roughness (p. 72 in "ANSYS
CFX-Solver Modeling Guide").
Heat Flux in theNear-WallRegion
The thermal boundary layer is modeled using the thermal law-of-the-wall function of B.A.
Kader [15].
Heat flux at the wall can be modeled using a wall function approach or the automatic wall
treatment. Using similar assumptions as those above, the non-dimensional near-wall
temperature profile follows a universal profile through the viscous sublayer and the
logarithmic region. The non-dimensional temperature, T+, is defined as:
(Eqn. 182)
where Tw is the temperature at the wall, Tf the near-wall fluid temperature, cp the fluid heat
capacity and qw the heat flux at the wall. The non-dimensional temperature distribution is
then modeled as:
(Eqn. 183)
where:
(Eqn. 184)
(Eqn. 185)
Pr is the fluid Prandtl number, given by:
(Eqn. 186)
where is the fluid thermal conductivity. Combining these equations leads to a simple
form for the wall heat flux model:
(Eqn. 187)
T+ ρcpu* Tw T– f( )qw
----------------------------------------=
T+ Pr y*e Γ–( ) 2.12 y*( )ln β+[ ]e 1– Γ⁄( )+=
β 3.85Pr1 3⁄ 1.3–( )2
2.12 Pr( )ln+=
Γ 0.01 Pr y*( )4
1 5Pr3 y*+-----------------------------=
Prµcpλ
--------=
λ
qwρcpu*
T+--------------- Tw T f–( )=
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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Turbulent fluid flow and heat transfer problems without conjugate heat transfer objects
require the specification of the wall heat flux, qw, or the wall temperature, Tw. The energy
balance for each boundary control volume is completed by multiplying the wall heat flux by
the surface area and adding to the corresponding boundary control volume energy
equation. If the wall temperature is specified, the wall heat flux is computed from the
equation above, multiplied by the surface area and added to the boundary energy control
volume equation.
AdditionalVariables
The treatment of additional scalar variables in the near wall region is similar to that for heat
flux.
Treatment ofCompressibilityEffects
With increasing Mach number (Ma > 3), the accuracy of the wall-functions approach
degrades, which can result in substantial errors in predicted shear stress, wall heat transfer
and wall temperature for supersonic flows.
It has been found that the incompressible law-of-the-wall is also applicable to compressible
flows if the velocity profile is transformed using a so-called ”Van Driest transformation” [16].
The logarithmic velocity profile is given by:
(Eqn. 188)
where , , = 0.41 and C = 5.2. The subscript w refers to
wall conditions, and the subscript “comp” refers to a velocity defined by the following
equation (transformation):
(Eqn. 189)
Near a solid wall, the integrated near-wall momentum equation reduces to:
(Eqn. 190)
while the energy equation reduces to:
(Eqn. 191)
Expressions for shear stress and heat flux applicable to the boundary layer region are:
(Eqn. 192)
U compuτ
---------------1κ--- u* y
v---------⎝ ⎠
⎛ ⎞ln C+=
uτ τw ρw⁄( )1 2⁄= y+ uτ y vw⁄( )= κ
U compρ
ρw------ Ud∫=
τ τw=
q qw U τw+=
τ µtU∂y∂
-------=
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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and
(Eqn. 193)
If (Eqn. 190), (Eqn. 192) and (Eqn. 193) are substituted into (Eqn. 191) and integrated, the
resulting equation is:
(Eqn. 194)
which provides a relationship between the temperature and velocity profiles. Using the
perfect gas law, and the fact that the pressure is constant across a boundary layer, (Eqn. 194)
replaces the density ratio found in (Eqn. 189). Performing the integration yields the
following equation for the “compressible” velocity:
(Eqn. 195)
where:
(Eqn. 196)
(Eqn. 197)
(Eqn. 198)
The above derivation provides most of the equations necessary for the implementation of
wall-functions which are applicable to compressible flows. These wall-functions are
primarily implemented in two locations in ANSYS CFX: hydrodynamics and energy. First,
consider the implementation in the hydrodynamics section. The equation for the
wall-shear-stress is obtained after a slight rearrangement of (Eqn. 188):
(Eqn. 199)
τ µtU∂y∂
-------=
T TwPrt qwU
cp τw--------------------–
Prt U 2
2cp----------------–=
U comp B A U+D
---------------⎝ ⎠⎛ ⎞ A
D----⎝ ⎠
⎛ ⎞asin–asin=
A qw τw⁄=
B 2cpTw( ) Prt( )⁄=
D A2 B+=
τwρ-----
u*1k-- u* y
v---------ln C+
-----------------------------U comp=
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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This is similar to the low speed wall-function, except that Ucomp now replaces U. The Van
Driest transformation given by (Eqn. 195), must now be performed on the near wall velocity.
In the implementation, (Eqn. 195) is re-written in the following form:
(Eqn. 200)
where:
(Eqn. 201)
This completes the wall-function modifications for the hydrodynamics.
The relationship between wall heat transfer, near wall velocity and near wall temperature is
given by (Eqn. 194) (and also in rearranged form after (Eqn. 200): the equation for Tw / T). A
dimensionless variable, , can be defined as:
(Eqn. 202)
and hence:
(Eqn. 203)
Ec( )comp1 2⁄ 2
Prt-------
TwT-------
1 2⁄ B* Ec+D*
------------------⎝ ⎠⎛ ⎞ B*
D*-------⎝ ⎠
⎛ ⎞asin–asin=
u* Cµ1 4⁄ k1 2⁄
=
τ*w ρ u*( )2=
Ec U 2
cp T---------=
B*qw
cp T τ*w-----------------U=
D* B*2 2Prt-------
TwT-------Ec+⎝ ⎠
⎛ ⎞ 1 2⁄=
U comp cpT Ec( )comp( )1 2⁄=
C 5.2=
Prt 0.9=
TwT------- 1 Prt B*
Prt2
------- Ec+ +=
θ+
θ+ PrtUu*-----=
θ+ PrtUu*-------
ρcp u*qw
---------------- Tw T–Prt U 2
2cp---------------–
⎝ ⎠⎜ ⎟⎛ ⎞
= =
Turbulence and Wall Function Theory: Modeling Flow Near the Wall
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From (Eqn. 203) it is clear that knowing , the near wall velocity, the near wall temperature
and the wall heat transfer, the wall temperature can be calculated (if instead the wall
temperature is known, then the wall heat transfer is calculated). Before proceeding, it is
instructive to consider the equation that is used in ANSYS CFX for low Mach number heat
transfer. The equation is subsequently modified for use at higher Mach numbers. This
unmodified equation is given by:
(Eqn. 204)
where:
(Eqn. 205)
(Eqn. 206)
(Eqn. 207)
(Eqn. 208)
It can be seen that this equation blends between the linear and logarithmic near wall
regions, and hence is more general than just using the logarithmic profile that is implied by
(Eqn. 202). (Eqn. 204) has been extended for use in the compressible flow regime by
interpreting the linear and logarithmic velocity profiles given above as “compressible” or
Van Driest transformed velocities:
(Eqn. 209)
(Eqn. 210)
This change is consistent with (Eqn. 188). The “untransformed” velocities required by (Eqn.
204) can be obtained by applying the inverse Van Driest velocity transformation to these
“compressible” velocities. The inverse transformation is given by:
(Eqn. 211)
θ+
θ+ Pr U 1+( ) e Γ– Prt U 2
+( ) e 1 Γ⁄–+=
U 1+ y+
=
U 2+ 1
Prt------- 2.12 y+( )ln β+( )=
Γ f 1 Pr y+( )=
β f 2 Pr( )=
U comp1+ y+
=
U comp2+ 1
Prt------- 2.12 y+( )ln β+( )=
U + 1R--- RU comp
+( )sin H 1 RU comp+( )cos–( )–=
Turbulence and Wall Function Theory: Wall Distance Formulation
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where:
(Eqn. 212)
(Eqn. 213)
and A and B are defined following (Eqn. 195) above. The reverse transformed velocities
obtained from (Eqn. 211) are substituted into (Eqn. 204) to obtain the value of . Either the
wall heat transfer or the wall temperature can then be obtained from (Eqn. 203).
Wall Distance Formulation
The wall distance is used in various functions that control the transition between near-wall
and freestream models. Its derivation is described here.
1D Illustration of Concept
Consider the 1D case of a horizontal surface, with the y direction normal to the surface:
Consider the following Poisson equation for a variable :
(Eqn. 214)
with a boundary condition of at the wall.
Integrate once:
(Eqn. 215)
Integrate again:
(Eqn. 216)
R u*( ) B( )⁄=
H A u*( )⁄=
θ+
wall
y
y=o
φ
y2
2
d
d φ 1–=
φ 0=
yddφ y– C1+=
φ 0.5 y2– C1 y C2+ +=
Turbulence and Wall Function Theory: Wall Distance Formulation
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Since at , you can deduce that . You also know from (Eqn. 215) that:
(Eqn. 217)
Substituting into (Eqn. 216) for and , you are left with a quadratic equation for in
terms of and :
(Eqn. 218)
Rearranging this gives:
(Eqn. 219)
where , , and .
Solving this quadratic and choosing the positive root gives:
(Eqn. 220)
Concept Generalized to 3D
(Eqn. 214) becomes a diffusion-only transport equation with a uniform source term of unity:
(Eqn. 221)
You can solve this equation with wall boundary conditions of .
You then evaluate (Eqn. 220), replacing with ( is always positive), and interpret
as the distance to the nearest wall, or “wall distance,” which gives the following
expression for wall distance:
(Eqn. 222)
Since is always positive, the wall distance is also always positive.
φ 0= y 0= C2 0=
C1 yddφ y+=
C1 C2 y
φyd
dφ
φ 0.5 y2–
yddφ y+⎝ ⎠
⎛ ⎞ y+=
a y2 by c+ + 0=
a 0.5= byd
dφ= c φ–=
yyd
dφ–yd
dφ⎝ ⎠⎛ ⎞
22φ++=
φ∇2 1–=
φ 0=
yddφ φ∇
yddφ
y
Wall Distance ∇φ– ∇φ 2 2φ++=
φ
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ANSYS CFX-Solver Theory Guide
GGI and MFR Theory
Introduction
This chapter describes:
• Interface Characteristics (p. 120)
• Numerics (p. 120)
This chapter provides an overview of the numerical methods used for GGI (general grid
interface) in ANSYS CFX.
A control surface approach is used to perform the connection across a GGI attachment or
periodic condition. A physically based intersection algorithm is employed to provide the
complete freedom to change the grid topology and physical distribution across the
interface.
A general intersection algorithm permits connections to be successfully made, even when
the resultant surfaces on either side of an interface do not physically “fit” together to form a
well defined physical connection. In addition, an automatic surface trimming function is
performed by the GGI algorithm, to account for mismatched surface extent. This means that
a GGI attachment or periodic condition can be successfully defined where the surface on
one side of the interface is larger (in extent) than the surface on the other side of the
interface. The interface is constructed between the overlapping regions of the two sides of
the interface.
Multiple Frames of Reference (MFR) allows the analysis of situations involving domains that
are rotating relative to one another. For ANSYS CFX, this feature focuses on the investigation
of rotor/stator interaction for rotating machinery using the Frozen Rotor frame change
model. Since MFR is based on the GGI technology, the most appropriate meshing style may
be used for each component in the analysis.
GGI and MFR Theory: Interface Characteristics
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Interface Characteristics
The numerical algorithms employed, as well as the control surface treatment of the
numerical fluxes across the interface, are designed and implemented in such a way as to
provide for maximum robustness and accuracy. The treatment of the interface fluxes is fully
implicit and fully conservative in mass, momentum, energy, scalars, etc. This means that the
multigrid solver can be applied directly, without any penalty in terms of robustness or
convergence rate, to problems involving GGI conditions. Each of the different types of GGI
interfaces have the following attributes:
1. Strict conservation is maintained across the interface, for all fluxes of all equations (afteraccounting for changes in pitch).
2. The interface treatment is fully implicit, so that the presence of an interface does notadversely affect overall solution convergence.
3. The interface is applicable to incompressible, subsonic, transonic and supersonic flowconditions, and all model options within ANSYS CFX (e.g., turbulence models,multiphase models, mixture models, CHT, reaction, etc.).
4. The interface accounts internally for pitch change by scaling up or down (as required)the local flows as they cross the interface, for the case of frame change interfaces.
5. Any number of GGI connection conditions are possible within a computational domain.
The surface fluxes along each side of the interface are discretized in terms of nodal
dependent variables, and in terms of control surface equations and control surface
variables.
Numerics
If the case is transient rotor-stator, then the current relative position of each side of a sliding
interface is first computed at the start of each timestep. Each frame change model then
proceeds by discretizing the surface fluxes along each side of the interface in terms of nodal
dependent variables, and in terms of control surface equations and control surface
variables. Each interface surface flow is discretized using the standard flux discretization
approach, but employs both the nodal dependent variables on the local side of the interface
and control surface variables on the interface region. Balance equations within the interface
region are generated for the interface variables from the flux contributions from both sides
of the interface. These equations are called control surface equations (different from control
volume equations) because they enforce a balance of surface flows over a given surface
area.
In detail, the GGI connection condition is implemented as follows:
1. Define regions within the interface where the fluxes must balance: control surfaces.Within each control surface, identify new dependent variables. These are calledinterface variables. For a Stage interface, the balance is across the entire interface in thedirection of rotation, with as many control surfaces perpendicular to the direction ofrotation as the grid permits. For all other interfaces, the control surface balance is at theresolution of the interface grid.
GGI and MFR Theory: Numerics
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2. Evaluate the fluxes at each interface location, by visiting all control volumes withsurfaces exposed to the interface. Evaluate the surface flows using the `standard'approach taken for all interior flux evaluations for advection, diffusion, pressure inmomentum, and mass flows. Use a combination of nodal dependent variables and theinterface variables in these evaluations. For example consider advection; if the flow isinto the interface control volume, the advected quantity is equated to the interfacevariable. If the flow is out of the interface control volume, the advected quantity isequated to the local nodal control volume variable. Below is a summary of all commonflux discretizations at the interface:
• Advection: Mass out is connected to the upstream (nodal) values, and mass in isconnected to upstream (control surface) values.
• Diffusion: A diffusion gradient is estimated using the regular shape function basedgradient coefficients, but all dependence of the gradient estimate on nodes on theinterface are changed to a dependence on interface variables.
• Pressure in momentum: Evaluated using local nodal and control surface pressuresand shape function interpolations.
• Local pressure gradient in mass re-distribution: This gradient is estimated using
the regular shape function based gradient coefficients, but all dependence of the
gradient estimate on nodal pressure on the interface is in terms of the interface
pressure variable.
3. When a face is in contact with more than one control surface balance equation,discretize the fluxes at each integration point location in terms of generic interfaceunknowns, evaluate the flux N times (where N is the number of control surfaces incontact with the face), each time using a different control surface variable and applyinga weighting factor to the flow based on an `exposed fraction' basis. Each partial flow isaccumulated in the control volume equation and in the relevant control surfaceequation.
4. Include each surface flow evaluation in two places: once in the interface control volumeequation, and once in the adjacent control surface equation. Once all interface surfaceshave been visited, the resulting equation set is as follows:
• All interface control volume equations are complete. Each equation has coefficientsto the usual neighboring nodal variables, as well as to interface variables.
• All control surface equations are now complete. Each equation has coefficients to
local interface variables as well as to nodal variables.
5. Solve the linear equation set to yield values of all nodal variables (from the control
volume equations) and all interface variables (from the control surface equations).
GGI and MFR Theory: Numerics
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ANSYS CFX-Solver Theory Guide
Multiphase Flow Theory
Introduction
The topic(s) in this section include:
• The Homogeneous and Inhomogeneous Models (p. 125)
• Hydrodynamic Equations (p. 128)
• Multicomponent Multiphase Flow (p. 131)
• Interphase Momentum Transfer Models (p. 131)
• Solid Particle Collision Models (p. 141)
• Interphase Heat Transfer (p. 147)
• Multiple Size Group (MUSIG) Model (p. 152)
• The Algebraic Slip Model (p. 159)
• Turbulence Modeling in Multiphase Flow (p. 163)
• Additional Variables in Multiphase Flow (p. 166)
• Sources in Multiphase Flow (p. 170)
• Interphase Mass Transfer (p. 170)
• Free Surface Flow (p. 184)
Two distinct multiphase flow models are available in ANSYS CFX, a Eulerian–Eulerian
multiphase model and a Lagrangian Particle Tracking multiphase model. Additional
information on the Lagrangian Particle Tracking model theory is available. For details, see
Particle Transport Theory (p. 187).
This section covers the Eulerian-Eulerianmodel theory. For details, see Multiphase Flow
Modeling (p. 155 in "ANSYS CFX-Solver Modeling Guide").
You should be familiar with the mathematical implementation of the single-phase flow,
before reading this section. For details, see Basic Solver Capability Theory (p. 1).
Multiphase Flow Theory: Multiphase Notation
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Multiphase Notation
In addition to the notation given here, review the list of symbols. For details, see List of
Symbols (p. 2).
Different phases of fluids are denoted using lowercase Greek letters , , , etc. In general,
a quantity subscribed with , , , etc., refers to the value of the quantity for that particular
phase. For example, the volume fraction of is denoted . Thus, the volume
occupied by phase in a small volume around a point of volume fraction is given by:
(Eqn. 1)
The total number of phases is . The volume fraction of each phase is denoted , where
.
It is important to distinguish between the material density and the effective density of a
fluid . The material density, , is the density of the fluid if it is the only phase present, i.e.,
the mass of per unit volume of . The effective density is then defined as:
(Eqn. 2)
This is the actual mass per unit volume of phase , given that phase only occupies a
fraction of the volume, i.e., the mass of per unit volume of the bulk fluid.
The mixture density is given by:
(Eqn. 3)
Multiphase Total Pressure
The total pressure in a multiphase simulation is defined as:
(Eqn. 4)
This definition is used for both incompressible and compressible flows, whereas single
phase flows treat total pressure differently depending on the simulation. For details, see
Total Pressure (p. 14).
α β γ
α β γ
α rα V α
α V rα
V α rαV=
NP rα
α 1 to NP=
α ρα
α α
ρα rαρα=
α α
α
ρm ραrαα∑=
ptot pstat12--r
αραUα
2
α∑+=
Multiphase Flow Theory: The Homogeneous and Inhomogeneous Models
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The Homogeneous and Inhomogeneous Models
Two different sub-models are available for Eulerian-Eulerian multiphase flow, the
homogeneous model and the inter-fluid transfer (inhomogeneous) model.
The Inhomogeneous Model
Interfacial AreaDensity
Interfacial transfer of momentum, heat and mass is directly dependent on the contact
surface area between the two phases. This is characterized by the interfacial area per unit
volume between phase and phase , known as the interfacial area density, . Note
that it has dimensions of one over length.
Interfacial transfer can be modeled using either the particle or mixture models. These
essentially provide different algebraic prescriptions for the interfacial area density.
The Particle ModelThe Particle model for interfacial transfer between two phases assumes that one of the
phases is continuous (phase ) and the other is dispersed (phase ). The surface area per
unit volume is then calculated by assuming that phase is present as spherical particles of
Mean Diameter . Using this model, the interphase contact area is:
(Eqn. 5)
This simple model is modified for robustness purposes in two ways:
• is clipped to a minimum volume fraction to ensure the area density does not go
exactly to zero.
• For large (i.e., when the assumption of being dispersed is invalid), the area density
is decreased to reflect the fact that it should lead to zero as tends to 1.
With these modifications, the area density for the particle model is implemented as
(Eqn. 6)
where
(Eqn. 7)
α β Aαβ
α β
β
dβ
Aαβ6rβdβ--------=
rβ
rβ β
rβ
Aαβ6rβ∗dβ
-----------=
rβ∗max rβ rmin,( ) if rβ rmax<( )
max1 rβ–
1 rmax–-------------------rmax rmin,( ) if rβ rmax>( )
⎩⎪⎨⎪⎧
=
Multiphase Flow Theory: The Homogeneous and Inhomogeneous Models
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By default, and take values of 0.8 and 10-7, respectively. In some cases, it may be
appropriate to use a different value for ; for example, increasing it to 10-3 provides a
very crude nucleation model for boiling a subcooled liquid. is controlled by the
parameter Minimum Volume Fraction for Area Density.
Non-dimensional interphase transfer coefficients may be correlated in terms of the particle
Reynolds number and the fluid Prandtl number. These are defined using the particle
mean diameter, and the continuous phase properties, as follows:
(Eqn. 8)
(Eqn. 9)
where , and are the viscosity, specific heat capacity and thermal conductivity of
the continuous phase .
The Mixture Model
This is a very simple model which treats both phases , symmetrically. The surface area
per unit volume is calculated from
(Eqn. 10)
where is an interfacial length scale, which you must specify.
By way of example, suppose you have oil-water flow in which you may have either water
droplets in continuous oil, or oil droplets in continuous water, in the limits ,
respectively. Then, a simple model for interfacial area density which has the correct behavior
in these two limits is given by:
(Eqn. 11)
rmax rmin
rmin
rmin
Reαβρα Uβ Uα– dβ
µα-------------------------------------=
PrαβµαCPα
λα----------------=
µα CPα λα
α
α β
Aαβrαrβdαβ----------=
dαβ
rα 0→ rβ 0→
Aαβ6rαrβ
rαdβ rβdα+-----------------------------= dαβ
rαdβ rβdα+
6-----------------------------=⇒
Multiphase Flow Theory: The Homogeneous and Inhomogeneous Models
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Non-dimensional interphase transfer coefficients may be correlated in terms of the mixture
Reynolds number and Prandtl number defined as follows:
(Eqn. 12)
(Eqn. 13)
where , , and are the density, viscosity, specific heat capacity and
thermal conductivity of the mixture respectively, defined by:
(Eqn. 14)
The Free Surface ModelThe free surface model attempts to resolve the interface between the fluids. If there are just
two phases in the simulation, the following equation is used for interfacial area density:
(Eqn. 15)
When more than two phases are present, this is generalized as follows:
(Eqn. 16)
The Homogeneous Model
In homogeneous multiphase flow, a common flow field is shared by all fluids, as well as
other relevant fields such as temperature and turbulence. This allows some simplifications
to be made to the multifluid model resulting in the homogeneous model.
For a given transport process, the homogeneous model assumes that the transported
quantities (with the exception of volume fraction) for that process are the same for all
phases, i.e.,
(Eqn. 17)
Since transported quantities are shared in homogeneous multiphase flow, it is sufficient to
solve for the shared fields using bulk transport equations rather than solving individual
phasic transport equations.
Reαβραβ Uβ Uα– dαβ
µαβ------------------------------------------=
PrαβµCPαβ
λαβ----------------=
ραβ µαβ CPαβ λαβ
ραβ rαρα rβρβ+=
µαβ rαµα rβµβ+=
Aαβ rα∇=
Aαβ2 r∇ α r∇ β
r∇ α r∇ β+-------------------------------=
φα φ= 1 α NP≤ ≤
Multiphase Flow Theory: Hydrodynamic Equations
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The bulk transport equations can be derived by summing the individual phasic transport
equations (Eqn. 191) over all phases to give a single transport equation for :
(Eqn. 18)
where:
(Eqn. 19)
The homogeneous model does not need to be applied consistently to all equations. For
example, the velocity field may be modeled as inhomogeneous, but coupled with a
homogeneous turbulence model. Alternatively, a homogeneous velocity field may be
coupled with inhomogeneous temperature fields. Homogeneous additional variables are
also available in ANSYS CFX.
Hydrodynamic Equations
The following is a summary of the equations of momentum and mass transfer for
inhomogeneous and homogeneous multiphase flow in ANSYS CFX. The equivalent for
single-phase flow is also available. For details, see Transport Equations (p. 23).
Inhomogeneous Hydrodynamic Equations
MomentumEquations
(Eqn. 20)
• describes momentum sources due to external body forces, and user defined
momentum sources. For details, see Sources (p. 43).
• describes the interfacial forces acting on phase due to the presence of other
phases. Additional information for the models available for interfacial forces is available.For details, see Interphase Momentum Transfer Models (p. 131).
φ
t∂∂ ρφ( ) ρUφ Γ φ∇–( )∇•+ S=
ρ rαραα 1=
N P
∑=
U 1ρ--- rαραUα
α 1=
N P
∑=
Γ rαΓαα 1=
N P
∑=
t∂∂ rαραUα( ) rα ραUα Uα⊗( )( )∇•+
rα pα∇– rαµα Uα∇ Uα∇( )T+( )( )∇•+=
Γαβ+ U β Γβα
+ Uα–( )β 1=
N p
∑ SMα Mα+ + +
SMα
Mα α
Multiphase Flow Theory: Hydrodynamic Equations
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• The term:
(Eqn. 21)
represents momentum transfer induced by interphase mass transfer. For details, seeInterphase Mass Transfer (p. 170).
• The above momentum equations are valid for fluid phases only. For dispersed solid
phases, additional terms are present representing additional stresses due to particle
collisions.
ContinuityEquations
(Eqn. 22)
• describes user specified mass sources. For details, see Sources (p. 43).
• is the mass flow rate per unit volume from phase to phase . This term only
occurs if interphase mass transfer takes place. For details, see Interphase Mass Transfer
(p. 170).
VolumeConservationEquation
This is simply the constraint that the volume fractions sum to unity:
(Eqn. 23)
This equation may also be combined with the phasic continuity equations to obtain a
transported volume conservation equation. Take (Eqn. 22), divide by phasic density, and
sum over all phases. This yields:
(Eqn. 24)
Interpreting this equation is simpler if you consider the special case of incompressible
phases with no sources, in which it simplifies to:
(Eqn. 25)
which requires the volume flows to have zero divergence. (Eqn. 24) is the volume continuity
equation solved by the ANSYS CFX-Solver.
Γαβ+ U β Γβα
+ Uα–( )
∂∂t----- rαρα( ) ∇ rαραUα( )•+ SMSα Γαβ
β 1=
N p
∑+=
SMSα
Γαβ β α
rαα 1=
N P
∑ 1=
1ρα------
ρ∂ αt∂
--------- ∇ rαραUα( )•+⎝ ⎠⎛ ⎞
α∑ 1
ρα------ SMSα Γαβ
β 1=
N P
∑+⎝ ⎠⎜ ⎟⎛ ⎞
α∑=
∇ rαUα( )•α∑ 0=
Multiphase Flow Theory: Hydrodynamic Equations
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PressureConstraint
The complete set of hydrodynamic equations represent equations in the
unknowns , , , , . You need more equations to close the system.
These are given by constraints on the pressure, namely that all phases share the same
pressure field:
(Eqn. 26)
Homogeneous Hydrodynamic Equations
MomentumEquations
The homogeneous model for momentum transport assumes:
(Eqn. 27)
and is given by (Eqn. 22) and the momentum equation:
(Eqn. 28)
where:
(Eqn. 29)
The following points should be noted:
• The inter-phase transfer terms have all cancelled out.
• This is essentially a single phase transport equation, with variable density and viscosity.
ContinuityEquations
The homogeneous continuity equation is the same as for full multiphase, (Eqn. 22), except
that is not phase specific.
VolumeConservationEquations
The homogeneous volume conservation equation is the same as for full multiphase, (Eqn.
23), except that is not phase specific.
PressureConstraint
The pressure constraint given above for full multiphase flow is also used for homogeneous
multiphase flow.
4NP 1+ 5NP
U α V α Wα rα pα NP 1–
pα p for all α 1,......,NP= =
Uα U= 1 α NP≤ ≤,
t∂∂ ρU( ) ρU U⊗ µ U∇ U∇( )T
+( )–( )∇•+ SM p∇–=
ρ rαραα 1=
N P
∑=
µ rαµαα 1=
N P
∑=
U
U
Multiphase Flow Theory: Multicomponent Multiphase Flow
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Multicomponent Multiphase Flow
In multiphase multicomponent flows, transport equations for the mass fractions of
components are assumed to take a similar form those used for single-phase
multicomponent flow:
(Eqn. 30)
You should note that the molecular diffusion coefficients are given by , where
is the Kinematic Diffusivity.
Source terms in multicomponent multiphase flow behave the same as multicomponent
mass sources, but on a per fluid basis. For details, see Sources (p. 43).
Interphase Momentum Transfer Models
The theory described in this section only applies to inhomogeneous multiphase flow. When
using the homogeneous model, momentum transfer between phases is assumed to be very
large.
Interphase momentum transfer, , occurs due to interfacial forces acting on each phase
, due to interaction with another phase . The total force on phase due to interaction
with other phases is denoted , and is given by:
(Eqn. 31)
Note that interfacial forces between two phases are equal and opposite, so the net
interfacial forces sum to zero:
(Eqn. 32)
The total interfacial force acting between two phases may arise from several independent
physical effects:
(Eqn. 33)
The forces indicated above respectively represent the interphase drag force, lift force, wall
lubrication force, virtual mass force, turbulence dispersion force and solids pressure force
(for dense solid particle phases only).
Y Aα
t∂∂ rαραY Aα( ) rα ραUαY Aα ραDAα Y Aα∇( )–( )( )∇•+ SAα=
ραDAα DAα
Mαβ
α β α
Mα
Mα Mαββ α≠∑=
Mαβ M– βα=( ) Mαα∑ 0=⇒
Mαβ MαβD Mαβ
L MαβLUB Mαβ
VM MαβTD MS ....+ + + + + +=
Multiphase Flow Theory: Interphase Momentum Transfer Models
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ANSYS CFX provides a wide range of physical models for these forces. These models are
described in the following places:
• Interphase Drag (p. 132)
• Lift Force (p. 139)
• Wall Lubrication Force (p. 140)
• Virtual Mass Force (p. 139)
• Interphase Turbulent Dispersion Force (p. 141)
• Solid Particle Collision Models (p. 141)
Interphase Drag
The following general form is used to model interphase drag force acting on phase due
to phase :
(Eqn. 34)
Note that and . Hence, the sum over all phases of all interphase
transfer terms is zero.
In this section, you will learn how the coefficients may be computed from a knowledge
of dimensionless drag coefficients. The range of models available for drag coefficients is also
described.
The total drag force is most conveniently expressed in terms of the dimensionless drag
coefficient:
(Eqn. 35)
where is the fluid density, is the relative speed, is the magnitude of the
drag force and is the projected area of the body in the direction of flow. The continuous
phase is denoted by and the dispersed phase is denoted by .
α
β
Mα cαβd( ) U β Uα–( )=
cαα 0= cαβ cβα=
cαβd( )
CDD
12--ρα U α Uβ–( )2A------------------------------------------=
ρ Uα U β–( ) D
A
α β
Multiphase Flow Theory: Interphase Momentum Transfer Models
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Interphase Drag for the Particle Model
For spherical particles, the coefficients may be derived analytically. The area of a single
particle projected in the flow direction, , and the volume of a single particle are
given by:
(Eqn. 36)
where is the mean diameter. The number of particles per unit volume, , is given by:
(Eqn. 37)
The drag exerted by a single particle on the continuous phase is:
(Eqn. 38)
Hence, the total drag per unit volume on the continuous phase is:
(Eqn. 39)
Comparing with the Momentum Equations for phase , where the drag force per unit
volume is:
(Eqn. 40)
you get:
(Eqn. 41)
which can be written as:
(Eqn. 42)
This is the form implemented in ANSYS CFX.
cαβd( )
Ap V p
Apπd2
4---------=
V pπd3
6---------=
d np
nprβV p-------
6rβ
πd3---------= =
Dp12--CDραAp U β Uα– U β Uα–( )=
Dαβ npDp34--
CDd
-------rβρα U β Uα– U β Uα–( )= =
α
Dαβ cαβd( ) U β Uα–( )=
cαβd( ) 3
4--
CDd
-------rβρα U β Uα–=
cαβd( ) CD
8-------Aαβρα U β Uα–=
Multiphase Flow Theory: Interphase Momentum Transfer Models
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The following section describes drag correlations specific to dispersed multiphase flow.
SparselyDistributedSolid Particles
At low particle Reynolds numbers (the viscous regime), the drag coefficient for flow past
spherical particles may be computed analytically. The result is Stokes’ law:
(Eqn. 43)
For particle Reynolds numbers, e.g.,(Eqn. 8), which are sufficiently large for inertial effects to
dominate viscous effects (the inertial or Newton’s regime), the drag coefficient becomes
independent of Reynolds number:
(Eqn. 44)
In the transitional region between the viscous and inertial regimes, for
spherical particles, both viscous and inertial effects are important. Hence, the drag
coefficient is a complex function of Reynolds number, which must be determined from
experiment.
This has been done in detail for spherical particles. Several empirical correlations are
available. The one available in ANSYS CFX is due to Schiller and Naumann (1933) [6]. It can
be written as follows:
Schiller Naumann Drag Model
(Eqn. 45)
ANSYS CFX modifies this to ensure the correct limiting behavior in the inertial regime by
taking:
(Eqn. 46)
DenselyDistributedSolid Particles
Densely Distributed Solid Particles: Wen Yu Drag Model
(Eqn. 47)
Note that this has the same functional form as the Schiller Naumann correlation, with a
modified particle Reynolds number, and a power law correction, both functions of the
continuous phase volume fraction .
You may also change the Volume Fraction Correction Exponent from its default value of
-1.65, if you wish.
CD24Re------ ,= Re 1«
CD 0.44,= 1000 Re 1 2x105–≤ ≤
0.1 Re 1000< <
CD24Re------ 1 0.15Re0.687
+( )=
CD max 24Re------ 1 0.15Re0.687
+( ) 0.44,⎝ ⎠⎛ ⎞=
CD rc1.65– max 24
Re′-------- 1 0.15Re'
0.687+( ) 0.44,⎝ ⎠
⎛ ⎞=
Re' rcRe=
rc
Multiphase Flow Theory: Interphase Momentum Transfer Models
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Densely Distributed Solid Particles: Gidaspow Drag Model
(Eqn. 48)
This uses the Wen Yu correlation for low solid volume fractions , and switches to
Ergun’s law for flow in a porous medium for larger solid volume fractions.
Note that this is discontinuous at the cross-over volume fraction. In order to avoid
subsequent numerical difficulties, ANSYS CFX modifies the original Gidaspow model by
linearly interpolating between the Wen Yu and Ergun correlations over the range
.
You may also change the Volume Fraction Correction Exponent of the Wen Yu part of the
correlation from its default value of -1.65, if you wish.
SparselyDistributedFluid Particles(drops andbubbles)
At sufficiently small particle Reynolds numbers (the viscous regime), fluid particles behave
in the same manner as solid spherical particles. Hence, the drag coefficient is well
approximated by the Schiller-Naumann correlation described above.
At larger particle Reynolds numbers, the inertial or distorted particle regime, surface tension
effects become important. The fluid particles become, at first, approximately ellipsoidal in
shape, and finally, spherical cap shaped.
In the spherical cap regime, the drag coefficient is well approximated by:
(Eqn. 49)
Several correlations are available for the distorted particle regime. ANSYS CFX uses the Ishii
Zuber and Grace and correlations.
Sparsely Distributed Fluid Particles: Ishii-Zuber Drag ModelIn the distorted particle regime, the drag coefficient is approximately constant,
independent of Reynolds number, but dependent on particle shape through the
dimensionless group known as the Eotvos number, which measures the ratio between
gravitational and surface tension forces:
(Eqn. 50)
Here, is the density difference between the phases, is the gravitational acceleration,
and is the surface tension coefficient.
CD CD Wen Yu( ),= rc 0.8>
cαβd( ) 150
1 rc–( )2µc
rcd p2
-------------------------- 74--
1 rc–( )ρc U c U d–
d p---------------------------------------------- ,+= rc 0.8<
rd 0.2<
0.7 rc 0.8< <
CD cap( ) 83--=
Eog∆ρd p
2
σ----------------=
∆ρ g
σ
Multiphase Flow Theory: Interphase Momentum Transfer Models
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The Ishii-Zuber correlation gives:
(Eqn. 51)
In this case, ANSYS CFX automatically takes into account the spherical particle and spherical
cap limits by setting:
(Eqn. 52)
The Ishii Zuber Model also automatically takes into account dense fluid particle effects. For
details, see Densely Distributed Fluid Particles (p. 137).
Sparsely Distributed Fluid Particles: Grace Drag ModelHere the drag coefficient in the distorted particle regime is given by:
(Eqn. 53)
where the terminal velocity is given by:
(Eqn. 54)
where:
(Eqn. 55)
and:
(Eqn. 56)
(Eqn. 57)
is the molecular viscosity of water at some reference
temperature and pressure.
CD ellipse( ) 23--Eo1 2⁄
=
CD dist( ) min CD ellipse( ) CD cap( ),( )=
CD max CD sphere( ) CD dist( ),( )=
CD ellipse( ) 43-- gd
U T2
-------∆ρρc-------=
U T
U Tµc
ρcd p-----------M 0.149– J 0.857–( )=
Mµc
4 g∆ρ
ρ2σ3---------------- Morton Number= =
J0.94H0.751 2 H 59.3≤<
3.42H0.441 H 59.3>⎝⎜⎛
=
H 43--EoM 0.149– µc
µref--------⎝ ⎠
⎛ ⎞ 0.14–=
µref 0.0009 kg m 1– s 1–=
Multiphase Flow Theory: Interphase Momentum Transfer Models
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In this case, ANSYS CFX automatically takes into account the spherical particle and spherical
cap limits by setting:
(Eqn. 58)
DenselyDistributedFluid Particles
Densely Distributed Fluid Particles: Ishii-Zuber Drag ModelThe Ishii Zuber drag laws automatically take into account dense particle effects. This is done
in different ways for different flow regimes.
In the viscous regime, where fluid particles may be assumed to be approximately spherical,
the Schiller Naumann correlation is modified using a mixture Reynolds number based on a
mixture viscosity.
Densely Distributed Fluid Particles: Dense Spherical Particle Regime (Ishii Zuber)
(Eqn. 59)
Here, is the user defined Maximum Packing value. This is defaulted to unity for a
dispersed fluid phase.
In the distorted particle regime, the Ishii Zuber modification takes the form of a multiplying
factor to the single particle drag coefficient.
Densely Distributed Fluid Particles: Dense Distorted Particle Regime (Ishii Zuber)
(Eqn. 60)
CD dist( ) min CD ellipse( ) CD cap( ),( )=
CD max CD sphere( ) CD dist( ),( )=
CD sphere( ) 24Rem---------- 1 0.15Rem
0.687+( )=
Remρc U d U c– d p
µm----------------------------------=
µmµc------- 1
rdrdm--------–⎝ ⎠
⎛ ⎞2.5rdmµ*–
=
µ*µd 0.4µc+
µd µc+-------------------------=
rdm
CD ellipse( ) E rd( )CD∞=
CD∞23--Eo1 2⁄
=
E rd( )1 17.67 f rd( )6 7⁄
+( )18.67 f rd( )
------------------------------------------------=
f rd( )µcµm------- 1 rd–( )1 2⁄
=
Multiphase Flow Theory: Interphase Momentum Transfer Models
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Densely Distributed Fluid Particles: Dense Spherical Cap Regime (Ishii Zuber)
(Eqn. 61)
The Ishii Zuber correlation, as implemented in ANSYS CFX, automatically selects flow regime
as follows:
Densely Distributed Fluid Particles: Automatic Regime Selection (Ishii Zuber):
(Eqn. 62)
Densely Distributed Fluid Particles: Grace Drag ModelThe Grace drag model, is formulated for flow past a single bubble. For details, see Sparsely
Distributed Fluid Particles (drops and bubbles) (p. 135).
For high bubble volume fractions, it may be modified using a simple power law correction:
(Eqn. 63)
Here, is the single bubble Grace drag coefficient. Advice on setting the exponent value
for the power law correction is available. For details, see Densely Distributed Fluid Particles:
Grace Drag Model (p. 167 in "ANSYS CFX-Solver Modeling Guide").
Interphase Drag for the Mixture Model
In the mixture model, a non-dimensional drag coefficient is defined as follows:
(Eqn. 64)
where is the total drag exerted by phase on phase per unit volume.
The mixture density is given by:
(Eqn. 65)
and the interfacial area per unit volume is given by:
(Eqn. 66)
CD cap( ) 1 rd–( )2CD∞=
CD∞83--=
CD CD sphere( )= if CD sphere( ) CD ellipse( )≥
CD min CD ellipse( ) CD cap( ),( )= if CD sphere( ) CD ellipse( )<
CD rcpCD∞=
CD∞
CD
Dαβ CDραβAαβ Uβ Uα– Uβ Uα–( )=
Dαβ β α
ραβ
ραβ rαρα rβρβ+=
Aαβ
Aαβrαrβdαβ----------=
Multiphase Flow Theory: Interphase Momentum Transfer Models
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where is a user-specified mixture length scale.
Interphase Drag for the Free Surface Model
In the free surface model, interphase drag is calculated in the same way as for the mixture
model (see “Interphase Drag for the Mixture Model (p. 138)”), except that the interfacial area
density is given by:
(Eqn. 67)
When more than two phases are present, this is generalized as follows:
(Eqn. 68)
[The Free Surface Model]
Lift Force
The lift force acts perpendicular to the direction of relative motion of the two phases.
ANSYS CFX contains a model for the shear-induced lift force acting on a dispersed phase in
the presence of a rotational continuous phase. This is given by:
(Eqn. 69)
where is a non-dimensional lift coefficient. In a rotating frame of reference with rotation
vector , the lift force is given by:
(Eqn. 70)
The current implementation permits a user-specified lift coefficient only. This may be a
constant, or a CEL expression.
Virtual Mass Force
The virtual mass force is proportional to relative phasic accelerations as follows:
(Eqn. 71)
dαβ
Aαβ rα∇=
Aαβ2 r∇ α r∇ β
r∇ α r∇ β+-------------------------------=
FcL F– d
L rdρcCL U d U c–( ) ωc×= =
ωc curl U c=
CL
Ω
FcL F– d
L rdρcCL U d* U c
*–( ) ωc
* 2Ω+( )×= =
ωc* curl U c
*=
FcVM F– d
VM rdρcCVMDdU d
Dt--------------
DcU cDt
-------------–⎝ ⎠⎛ ⎞= =
Multiphase Flow Theory: Interphase Momentum Transfer Models
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In a rotating frame of reference with rotation vector , the virtual mass force in terms of
is modified by Coriolis theorem, and is given by:
(Eqn. 72)
The non-dimensional virtual mass coefficient for inviscid flow around an
isolated sphere. In general, depends on shape and particle concentration . may
be specified by the user as a constant or a CEL expression.
Wall Lubrication Force
Under certain circumstances, for example, bubbly upflow in a vertical pipe, the dispersed
phase is observed to concentrate in a region close to the wall, but not immediately adjacent
to the wall. This effect may be modeled by the wall lubrication force, which tends to push
the dispersed phase away from the wall. ANSYS CFX contains the model of Antal [88].
(Eqn. 73)
Here, is the relative velocity between phases, is the disperse phase
mean diameter, is the distance to the nearest wall, and is the unit normal pointing
away from the wall. Hence, the force acts to push the disperse phase away from the wall. The
coefficients were determined by numerical experimentation, , for
a sphere. So the force is only active in a thin layer adjacent to the wall:
(Eqn. 74)
Hence, this force will only be activated on a sufficiently fine mesh, and grid convergence can
only be expected on extremely fine meshes.
Note that these default coefficients are rather different from those recommended in the
original paper of Antal et al [88]. More recent numerical experiments by Krepper and Prasser
[89] obtained the values ,
Ω U *
FcVM F– d
VM rdρcCVMDd
*U d*
Dt--------------
Dc*U c
*
Dt-------------– 2Ω U d
* U c*
–( )×+⎝ ⎠⎜ ⎟⎛ ⎞
= =
CVM 0.5=
CVM CVM
FcLUB F– d
LUB r– dρcU r U r nw⋅( )nw–( )( )2
d p------------------------------------------------------max C1 C2
d pyw------+ 0,⎝ ⎠
⎛ ⎞nw= =
U r U c U d–= d p
yw nw
C1 0.01–= C2 0.05=
ywd p------
C1C2------ 5≈≤
C1 0.0064–= C2 0.016=
Multiphase Flow Theory: Solid Particle Collision Models
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Interphase Turbulent Dispersion Force
Favre AveragedDrag Model
ANSYS CFX implements a model for turbulent dispersion force, based on the Favre average
of the interphase drag force [90].
(Eqn. 75)
Here, is the momentum transfer coefficient for the interphase drag force. Hence, the
model clearly depends on the details of the drag correlation used . is the turbulent
Schmidt number for continuous phase volume fraction, currently taken to be .
is a user-modifiable CEL multiplier. Its default value is unity.
Lopez deBertodanoModel
The model of Lopez de Bertodano (1991) [20] was one of the first models for the turbulent
dispersion force:
(Eqn. 76)
Unfortunately, it is not possible to recommend universal values of for this model.
values of 0.1 to 0.5 have been used successfully for bubbly flow with bubble diameters of
order a few millimetres. However, values up to 500 have been required for other situations.
See Lopez de Bertodano [21] and Moraga et al [91].
This model is included in ANSYS CFX for historical back compatibility with ANSYS CFX.
However, the more universal Favre Averaged Drag model is recommended for all situations
where an appropriate value of is unknown.
Solid Particle Collision Models
This section presents the theoretical background of the solid particle collision models
implemented in ANSYS CFX.
The following topic(s) will be discussed:
• Solids Stress Tensor (p. 142)
• Solids Pressure (p. 142)
• Solids Bulk Viscosity (p. 143)
• Solids Shear Viscosity (p. 144)
• Granular Temperature (p. 145)
FcTD F– d
TD CTDCcd
νtcσtc-------
rd∇rd
---------rc∇
rc--------–⎝ ⎠
⎛ ⎞= =
Ccd
σtc
0.9
CTD
McTD M– d
TD CTDρckc rc∇–= =
CTD CTD
CTD
Multiphase Flow Theory: Solid Particle Collision Models
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Solids Stress Tensor
Additional stresses due to inter-particle collisions are modeled using a collisional solids
stress tensor in the solid phase momentum equation only:
(Eqn. 77)
Here, denotes solids pressure, denotes solids shear viscosity, and denotes solids
bulk viscosity. There are two important classes of models for these quantities:
EmpiricalConstitutiveEquations
There exist wide classes of models where the constitutive elements of the solids stress are
specified using empirical constitutive relations. See, for example, (Enwald et al [[97]]). In
many of these, the solids pressure, shear and bulk viscosities are simple functions of the
solid phase volume fraction.
Kinetic TheoryModels
These are a class of models, based on the kinetic theory of gases, generalized to take into
account inelastic particle collisions. In these models, the constitutive elements of the solids
stress are functions of the solid phase granular temperature, defined to be proportional to
the mean square of fluctuating solid phase velocity due to inter-particle collisions:
(Eqn. 78)
In the most general kinetic theory models, the granular temperature is determined from a
transport equation. However, in many circumstances, it is possible to ignore the transport
terms, and determine granular temperature from the resulting algebraic equation.
Solids Pressure
EmpiricalConstitutiveEquations
The most popular constitutive equations for solids pressure are due to (Gidaspow [18]).
These actually specify the solids pressure gradient, rather than solids pressure directly:
(Eqn. 79)
(Eqn. 80)
Where is the Elasticity Modulus, is the Reference Elasticity Modulus, is the
Compaction Modulus, and is the Maximum Packing Parameter.
τsij Psδij– µsU i∂x j∂
---------U j∂xi∂
--------- 23--–
U k∂xk∂
---------δij+⎝ ⎠⎛ ⎞ ζs
U k∂xk∂
---------δij+ +=
Ps µs ζs
Θs13--u's
2=
Ps Ps rs( ) ∇Ps⇒ G rs( )∇rs= =
G rs( ) G0ec rs rsm–( )
=
G rs( ) G0 c
rsm
Multiphase Flow Theory: Solid Particle Collision Models
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The Gidaspow model is implemented with an option for specifying the Reference Elasticity
Modulus and Compaction Modulus. There is also an option to specify the Elasticity
Modulus directly. There is also an option to specify the solids pressure directly. This
permits more general constitutive relations than those where the solids pressure is a
function of volume fraction only.
Kinetic TheoryModels
The kinetic theory model for solids pressure is similar to the equation of state for ideal gases,
modified to take account of inelastic collisions, and maximum solid packing.
(Eqn. 81)
Here, denotes the coefficient of restitution for solid-solid collisions, and denotes
the radial distribution function. Popular models for the radial distribution function are given
by:
Gidaspow (1994) [18]:
(Eqn. 82)
Lun and Savage (1986) [100]:
(Eqn. 83)
Note that the radial distribution function tends to infinity as . The singularity is
removed in ANSYS CFX by setting:
(Eqn. 84)
where and:
(Eqn. 85)
Solids Bulk Viscosity
ConstitutiveEquationModels
Most simple constitutive equation models ignore the solids bulk viscosity. However, it is
possible for the user to specify a model for this, using a CEL expression, if required. Note that
any solids volume fraction dependence must be included in the CEL expression.
G rs( )
Ps ρsrsΘs 1 2 1 e+( )g0rs+( )=
e g0 rs( )
g0 rs( ) 0.6 1 rs rsm⁄( )1 3⁄–( )
1–=
g0 rs( ) 1 rs rsm⁄( )–( )2.5rsm–
=
rs rsm→
g0 rs( ) C0 C1 rs rc–( ) C2 rs rc–( )2 C3 rs rc–( )3+ + += r rc≥( ),
rc rsm 0.001–=
C0 1079=
C1 1.08 106×=
C2 1.08 109×=
C3 1.08 1012×=
Multiphase Flow Theory: Solid Particle Collision Models
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Kinetic TheoryModels
There is general agreement on the correct kinetic theory form for the solids bulk viscosity
(Lun et al 1984 [99]).
(Eqn. 86)
Solids Shear Viscosity
ConstitutiveEquationModels
The simplest constitutive equation model for solids shear viscosity was presented by Miller
and Gidaspow (1992) [106]. They successfully modeled gas-solid flow in a riser using a solids
shear viscosity linearly proportional to the solids phase volume fraction.
(Eqn. 87)
Note that their constant of proportionality is dimensional, and is likely to require
modification for different fluid-solid material properties.
More complex models for solids shear stress allow the shear stress to become very large in
the limit of maximum packing. A wide range of such models is discussed in the review article
by Enwald et al [97].
It is possible for the user to implement any of these models, using a CEL expression for the
solids shear viscosity. Note that any solids volume fraction dependence must be included in
the CEL expression.
Kinetic TheoryModels
Typically, the shear viscosity is expressed as a sum of at least two contributions: the kinetic
and collisional contributions:
(Eqn. 88)
There is wide agreement on the correct form of the collisional contribution. As in the kinetic
theory of gases, it is proportional to the square root of the granular temperature:
(Eqn. 89)
However, there are many proposals in the literature for the correct form of the kinetic
contribution. For example:
Gidaspow (1994) [18]:
(Eqn. 90)
ζs43--rs
2ρsd p g0 1 e+( ) Θπ----=
µs 5.35 Poise[ ] r× s=
µs µs col, µs kin,+=
µs col,45--rs
2ρsd p g0 1 e+( ) Θπ----=
µs kin,5 π48
----------ρsd p
1 e+( )g0---------------------- 1 4
5-- 1 e+( )g0rs+⎝ ⎠
⎛ ⎞ 2Θ=
Multiphase Flow Theory: Solid Particle Collision Models
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Lun and Savage (1986) [100]:
(Eqn. 91)
Here, .
Kinetic contributions are omitted from ANSYS CFX.
Granular Temperature
ANSYS CFX is restricted to models where the granular temperature is determined
algebraically.
AlgebraicEquilibriumModel
may be specified directly by the user, or from the assumption of local equilibrium in a
transport equation model. The latter is based on:
(Eqn. 92)
where denotes the solids shear stress tensor (Eqn. 77), and:
(Eqn. 93)
Expand the production term:
(Eqn. 94)
(Eqn. 95)
where:
(Eqn. 96)
µs kin,5 π96
----------ρsd p1
ηg0---------
85--rs+⎝ ⎠
⎛ ⎞1 8
5--η 3η 2–( )rs g0+
2 η–-----------------------------------------------
⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞
Θ=
η 12-- 1 e+( )=
Θs
Θs
Production Dissipation τsijU i∂x j∂
---------⇒ γ s= =
τsij
γ s 3 1 e2–( )rs
2ρs g0Θs4ds----
Θsπ------
U k∂xk∂
---------–⎝ ⎠⎜ ⎟⎛ ⎞
⎝ ⎠⎜ ⎟⎛ ⎞
=
τsijU i∂x j∂
--------- PsU k∂xk∂
---------– µsU i∂x j∂
---------U j∂xi∂
---------+⎝ ⎠⎛ ⎞ U i∂
x j∂--------- ζs
23--µs–⎝ ⎠
⎛ ⎞ U k∂xk∂
---------⎝ ⎠⎛ ⎞ 2
+ +=
P⇒ roduction PsD– µsS2 λsD
2+ +=
λs ζs23--µs–= D
U k∂xk∂
---------= S2 12--
U i∂x j∂
---------U j∂xi∂
---------+⎝ ⎠⎛ ⎞ 2
=
Multiphase Flow Theory: Solid Particle Collision Models
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In order to determine from (Eqn. 92), it is useful to take into account the dependence of
solids pressure and shear bulk viscosities on . You have:
(Eqn. 97)
so you may write:
(Eqn. 98)
(Eqn. 99)
Hence, substituting into (Eqn. 95), you may express the production term in terms of granular
temperature as follows:
(Eqn. 100)
where:
(Eqn. 101)
Similarly, the dissipation term (Eqn. 93) may be simplified as follows:
(Eqn. 102)
where:
(Eqn. 103)
Equating (Eqn. 100) and (Eqn. 102), and dividing by gives a quadratic equation for :
(Eqn. 104)
Θs
Θs
Ps Θs∝ µs ζs Θs1 2⁄∝,
Ps Ps0( )Θs= µs µs
0( )Θs1 2⁄
=
ζs ζs0( )Θs
1 2⁄= λs λs
0( )Θs1 2⁄
=
Production λs0( )D2 µs
0( )S2+( )Θs
1 2⁄ Ps0( )DΘs–=
APΘs1 2⁄ BPΘs–=
AP λs0( )D2 µs
0( )S2 0≥+= BP Ps0( )D=
Dissipation ADΘs3 2⁄ BDΘs–=
AD4
ds π------------CD 0≥=
BD CDD=
CD 3 1 e2–( )rs
2ρs g0 0≥=
Θs1 2⁄ Θs
ADΘs BP BD–( )Θs1 2⁄ AP–+ 0=
Multiphase Flow Theory: Interphase Heat Transfer
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Note that is strictly positive if, and only if, the coefficient of restitution is strictly less than
unity. In this case, in view of the fact that , (Eqn. 104) has a unique positive solution:
(Eqn. 105)
The Algebraic Equilibrium model has the flaw that unphysically large granular temperatures
can be generated in regions of low solid particle volume fraction. To circumvent this, it is
recommended that you specify an upper bound for the granular temperature. The square
of the mean velocity scale is a reasonable estimate for this.
Zero EquationModel
The Zero Equation Model implements the simpler algebraic model of Ding and Gidaspow
[98].
(Eqn. 106)
Interphase Heat Transfer
In the multiphase model, there are separate enthalpy and temperature fields for each phase.
Phasic Equations
Heat transfer is governed by the multiphase energy equations, which generalize the single
phase energy equations (Eqn. 6) and (Eqn. 7).
The multiphase version of the Total Energy equation is:
(Eqn. 107)
AD
AP 0≥
Θs1 2⁄ BD BP– BD BP–( )2 4ADAP++
2AD--------------------------------------------------------------------------------=
Θs1
15 1 e–( )----------------------ds
2S2=
t∂∂ rαραhα tot,( ) rα
p∂t∂
------ rαραUαhα tot, rαλα∇Tα–( )∇•+–
rαµα ∇Uα ∇Uα( )T 23--∇Uαδ–+⎝ ⎠
⎛ ⎞Uα⎝ ⎠⎛ ⎞∇•–
Γαβ+ hβ tot, Γβa
+ hα tot,–( )β 1=
N p
∑ Qα Sα+ +=
Multiphase Flow Theory: Interphase Heat Transfer
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The multiphase version of the thermal energy equation for static enthalpy (incompressible
and low speed compressible flows only) is:
(Eqn. 108)
where:
• , , , denote the static enthalpy, the temperature, and the thermal conductivity
of phase .
• describes external heat sources. For details, see Sources (p. 43).
• denotes interphase heat transfer to phase across interfaces with other phases.
For details, see Inhomogeneous Interphase Heat Transfer Models (p. 148).
• The term
(Eqn. 109)
represents heat transfer induced by interphase mass transfer. For details, see Interphase
Mass Transfer (p. 170).
Inhomogeneous Interphase Heat Transfer Models
Interphase heat transfer occurs due to thermal non-equilibrium across phase interfaces. The
total heat per unit volume transferred to phase due to interaction with other phases is
denoted , and is given by:
(Eqn. 110)
where:
(Eqn. 111)
t∂∂ rαρhα( ) rα ραUαhα λα Tα∇–( )( )∇•+
Γαβ+ hβs Γβa
+ hαs–( )β 1=
N p
∑ Qα Sα+ +=
hα Tα λα
α
Sα
Qα α
Γαβ+ hβs Γβa
+ hαs–( )
α
Qα
Qα Qαββ α≠∑=
Qαβ Q– βα= Qαα∑ 0=⇒
Multiphase Flow Theory: Interphase Heat Transfer
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Overall HeatTransferCoefficients
Heat transfer across a phase boundary is usually described in terms of an overall heat
transfer coefficient , which is the amount of heat energy crossing a unit area per unit
time per unit temperature difference between the phases. Thus, the rate of heat transfer,
, per unit time across a phase boundary of interfacial area per unit volume , from
phase to phase , is:
(Eqn. 112)
This may be written in a form analogous to momentum transfer:
(Eqn. 113)
where the volumetric heat transfer coefficient, , is modeled using the correlations
described below.
Particle ModelCorrelations
For particle model, the volumetric heat transfer coefficient is modeled as:
(Eqn. 114)
Hence, the interfacial area per unit volume and the heat transfer coefficient are
required.
More information on interfacial area density calculation is available.
It is often convenient to express the heat transfer coefficient in terms of a dimensionless
Nusselt number:
(Eqn. 115)
In the particle model, the thermal conductivity scale is taken to be the thermal
conductivity of the continuous phase, and the length scale is taken to be the mean
diameter of the dispersed phase:
(Eqn. 116)
For laminar forced convection around a spherical particle, theoretical analysis shows that
. For a particle in a moving incompressible Newtonian fluid, the Nusselt number is
a function of the particle Reynolds number and the surrounding fluid Prandtl number
.
hαβ
Qαβ Aαβ
β α
Qαβ hαβAαβ Tβ Tα–( )=
Qαβ cαβh( ) Tβ Tα–( )=
cαβh( )
cαβh( ) hαβAαβ=
hαβ
h λNud
-----------=
λ
d
hαβλαNuαβ
dβ--------------------=
Nu 2=
Re
Pr µαCPα λα⁄=
Multiphase Flow Theory: Interphase Heat Transfer
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Additional information on models in ANSYS CFX is available. For details, see Particle Model
Correlations for Overall Heat Transfer Coefficient (p. 172 in "ANSYS CFX-Solver Modeling
Guide"). Some additional details for the Interface Flux model are provided below.
• Interface Flux
The heat flux coefficients for both fluids and the interfacial heat flux value, F12, from
Fluid 1 to Fluid 2 (Fluid 1 is the fluid to appear on the left of the Fluid Pairs list) arespecified. F12 is the rate of heat transfer per unit time per unit interfacial area from phase
1 to phase 2. Hence, the heat transferred to fluid 2 from fluid 1 per unit volume is givenby:
(Eqn. 117)
may be given as a constant or an expression.
Typically, will be a function of the fluid 1 and fluid 2 temperature fields, and possibly
other variables. In this case, the user may accelerate convergence of the coupled solverby also specifying optional fluid 1 and fluid 2 heat flux coefficients.
(Eqn. 118)
The solver takes the absolute value of these flux coefficients to ensure that they arepositive. This is required for numerical stability. The partial derivatives need not becomputed exactly; it is sufficient for the specified coefficients to simply approximate thepartial derivatives. Specification of heat flux coefficients only affects the convergencerate to the solution of the coupled heat transfer equations, it does not affect theaccuracy of the converged solution.
For example, the simple model using a heat transfer coefficient multiplied by a bulk
temperature difference my be recovered using:
Mixture ModelCorrelations
When using the mixture model, the Nusselt number is defined in terms of a mixture
conductivity scale and the mixture length scale:
(Eqn. 119)
For details, see Mixture Model Correlations for Overall Heat Transfer Coefficient (p. 173 in
"ANSYS CFX-Solver Modeling Guide").
The TwoResistanceModel
There are special situations where the use of an overall heat transfer coefficient is not
sufficient to model the interphase heat transfer process. A more general class of models
considers separate heat transfer processes either side of the phase interface. This is achieved
by using two heat transfer coefficients defined on each side of the phase interface.
Q21 Q12– A12F12= =
F12
F12
h1∂F12∂T1----------- 0≥ ,≈ h2
∂F21∂T2-----------≈ 0≥
F12 F21– h T1 T2–( ),= = h1 h2 h= =
hαβλαβNuαβ
dαβ---------------------- ,= λαβ rαλα rβλβ+=
Multiphase Flow Theory: Interphase Heat Transfer
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Defining the sensible heat flux to phase from the interface as:
(Eqn. 120)
and the sensible heat flux to phase from the interface as:
(Eqn. 121)
where and are the phase and phase heat transfer coefficients respectively.
is interfacial temperature, and it is assumed to be the same for both phases.
The fluid-specific Nusselt number is defined as:
(Eqn. 122)
where is the thermal conductivity of fluid , and is the interfacial length scale (the
mean particle diameter for the Particle Model, and the mixture length scale for the Mixture
Model).
In the absence of interphase mass transfer, you must have overall heat balance
. This condition determines the interfacial temperature:
(Eqn. 123)
It also determines the interphase heat fluxes in terns of an overall heat transfer coefficient:
(Eqn. 124)
Hence, in the absence of interphase mass transfer, the two resistance model is somewhat
superfluous, as it may be implemented using a user-specified overall heat transfer
coefficient.
It is possible to specify a zero resistance condition on one side of the phase interface. This is
equivalent to an infinite fluid specific heat transfer coefficient . Its effect is to force
the interfacial temperature to be the same as the phase temperature, .
Modeling advice is available. For details, see Two Resistance Model for Fluid Specific Heat
Transfer Coefficients (p. 173 in "ANSYS CFX-Solver Modeling Guide").
α
qα hα Ts Tα–( )=
β
qβ hβ Ts Tβ–( )=
hα hβ α β Ts
Nuαhαdαβ
λα---------------=
λα α dαβ
qα qβ+ 0=
TshαTα hβTβ+
hα hβ+--------------------------------=
qα qβ– hαβ Tβ Tα–( ),= = 1hαβ-------- 1
hα------ 1
hβ-----+=
hα ∞→
Ts Tα=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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Homogeneous Heat Transfer in Multiphase Flow
For all transport processes other than heat transfer, the shared field is the same as the
transported quantity in the equation. However, in the case of heat transfer, it is temperature
which is shared but enthalpy which is transported. Hence, ANSYS CFX does not solve a bulk
enthalpy equation, but rather solves a separate enthalpy equation for each fluid with a large
interphase heat transfer term which forces the temperature fields to be the same.
The equations solved are identical to the phasic equations for full multiphase described
above. For homogeneous heat transfer model, the interphase heat transfer coefficient is not
modeled by any of the correlations used in full multiphase. Instead it is chosen to be large
relative to the other transported processes in the equation, thereby ensuring the phasic
temperatures are the same.
Multiple Size Group (MUSIG) Model
The MUSIG (Multiple Size Group) model has been developed to handle polydispersed
multiphase flows. By polydispersed, it means that the dispersed phase has a large variation
in size. One of the attributes of polydispersed multiphase flow is that the different sizes of
the dispersed phases interact with each other through the mechanisms of breakup and
coalescence. Population balance is a well-established method for calculating the size
distribution of a polydispersed phase, including breakup and coalescence effects. MUSIG
provides a framework in which the population balance method can be incorporated into
three-dimensional CFD calculations.
Model Derivation
PopulationBalanceEquation
The starting point for the MUSIG model is the population balance equation which
represents the continuity of particles of size . Let represent the number density of
particles of size at time . Then the population balance equation is:
(Eqn. 125)
v n v t,( )
v t
t∂∂ n v t,( )
xi∂∂ U i v t,( )n v t,( )( )+ BB DB– BC DC–+=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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where , , , and respectively represent the birth rate due to breakup of larger
particles, the death rate due to breakup into smaller particles, the birth rate due to
coalescence of smaller particles, and the death rate due to coalescence with other particles.
These rates may further be expressed as:
(Eqn. 126)
(Eqn. 127)
(Eqn. 128)
(Eqn. 129)
where represents the specific breakup rate (the rate at which particles of size
break into particles of size and ) and represents the specific coalescence
rate (the rate at which particles of size coalesce with particles of size to form particles of
size .
Size FractionEquations
The next step of the MUSIG model is to discretize (Eqn. 125) into size groups, or bins. Let
represent the number density of size group :
(Eqn. 130)
Also define the mass and volume fraction of size group to be and , respectively, and
recognize that . Now integrate (Eqn. 125) over the bin size dimension and
multiply by to give:
(Eqn. 131)
or:
(Eqn. 132)
BB DB BC DC
BB g ε v;( )n ε t,( ) εdv∞∫=
DB n v t,( ) g v ε;( ) εd0v∫=
BC12-- Q v ε ε;–( )n v ε t,–( )n v t,( ) εd0
v∫=
DC n v t,( ) Q v ε;( )n ε t,( ) td0∞∫=
g v ε;( ) v
ε v ε– Q v ε;( )
v ε
v ε+
Ni
i
Ni t( ) n v t,( ) vdvi 1 2⁄–
vi 1 2⁄+∫=
i mi ri
miNi ρiri=
mi
miNi t( )( )∂t∂
---------------------------xi∂∂ U i
i t( )Ni t( )( )+ Si=
ρiri( )∂t∂
----------------xi∂∂ ρiriU i
i( )+ Si=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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Defining the size fraction , this equation may also be written as:
(Eqn. 133)
which is the size fraction equation used by the MUSIG model. A further simplification is to
assume that all size groups share the same density and velocity yielding the
homogeneous MUSIG model:
(Eqn. 134)
Source Terms The contribution of the birth rate due to breakup of larger particles to the source term in
(Eqn. 134) is:
(Eqn. 135)
Similarly, the contribution of the death rate due to breakup into smaller particles is:
(Eqn. 136)
Note that the total source due to breakup is zero when summed over all size groups:
(Eqn. 137)
For the discretized coalescence sources, you must define the coalescence mass matrix
as the fraction of mass due to coalescence between groups at which goes into group :
(Eqn. 138)
f i ri rd⁄=
t∂∂ ρird f i( )
xi∂∂ ρirdU i
i f i( )+ Si=
ρd udi
t∂∂ ρdrd f i( )
xi∂∂ ρdrdU d
i f i( )+ Si=
BBi mi BB vdvi 1 2⁄–
vi 1 2⁄+∫=
mi g v j vi;( )N jj i>∑=
ρdrd g v j vi;( ) f jj i>∑⎝ ⎠
⎛ ⎞=
DBi ρdrd f i g vi v j;( )j 1<∑⎝ ⎠
⎛ ⎞=
BBi DBi–( )i
∑ 0=
X jki
j t i
X jki
m j mk+( ) mi 1––
mi mi 1––------------------------------------------- if mi 1– m< j mk mi<+
mi 1+ m j m+ k( )–
mi 1+ m– i--------------------------------------------
0
if mi m< j mk mi 1+<+
otherwise⎩⎪⎪⎪⎨⎪⎪⎪⎧
=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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The contribution of the birth rate due to coalescence of smaller particles to the source term
in (Eqn. 134) is:
(Eqn. 139)
Similarly, the contribution of the death rate due to coalescence into larger groups is:
(Eqn. 140)
Note that this formulation for the coalescence source terms guarantees that the total source
to coalescence is zero when summed over all size groups:
(Eqn. 141)
This follows from the requirement that together with the following
property of the mass matrix for all :
(Eqn. 142)
Size Group Discretization
The size groups may be discretized using one of three recipes. In each case, a minimum and
maximum diameter for the polydispersed fluid must be set along with the number of size
groups. In all cases, the diameter and mass of a particular group are related by:
(Eqn. 143)
If the fluid density is not constant, a constant reference density must be used in this
expression to ensure that the group sizes are independent of local density.
BCi mi BC vdvi 1 2⁄–
vi 1 2⁄+∫=
mi12-- Q v j vk;( )X jkiN jNk
k i≤∑
j i≤∑⎝ ⎠
⎛ ⎞=
ρdrd( )2 12-- Q v j vk;( )X jki f j f k
m j mk+
m jmk--------------------
k i≤∑
j i≤∑⎝ ⎠
⎛ ⎞=
DCi ρdrd( )2 Q vi v j;( ) f i f j1
m j------
j∑⎝ ⎠
⎛ ⎞=
BCi DCi–( )i
∑ 0=
Q vi v j;( ) Q vd vi;( )=
j k,
X jkii
∑ 1=
m π6---ρ
dd3
=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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Equal MassDiscretization
In this case, the minimum and maximum mass are calculated from the minimum and
maximum diameter using (Eqn. 143). The mass of group is calculated using:
(Eqn. 144)
(Eqn. 145)
Equal diameterdiscretization
In this case, the diameter of group is calculated from:
(Eqn. 146)
(Eqn. 147)
and the group mass is calculated from (Eqn. 143).
Geometric MassDiscretization
In this case, a geometrical progression for mass as follows:
(Eqn. 148)
Comparison The following table compares diameters for the three discretization options for a
polydispersed fluid having and .
The equal mass distribution resolves large bubbles the most, the geometric distribution
resolves the small bubbles well, and the equal diameter distribution weights both equally.
i
mi mmin ∆m i 12--–⎝ ⎠
⎛ ⎞+=
∆mmmax mmin–
N--------------------------------=
i
di dmin ∆d i 12--–⎝ ⎠
⎛ ⎞+=
∆ddmax dmin–
N----------------------------=
mi mmin34--
mmax mmin–
2N i–--------------------------------+=
Group Equal Mass Equal Diameter Geometric Mass
1 585 50 22.5
2 843 150 28.4
3 1000 250 35.8
4 1118 350 45.1
5 1216 450 56.8
6 1300 550 71.5
7 1375 650 90.1
8 1442 750 114
9 1503 850 143
10 1560 950 180
11 1613 1050 227
12 1663 1150 286
13 1710 1250 360
dmin 0mm= dmax 2mm=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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Breakup Models
Breakup kernels are often expressed as a function of the breakup fraction:
(Eqn. 149)
ANSYS CFX-Solver supports two breakup models.
Luo andSvendsen Model
Luo and Svendsen [61] developed a theoretical model for the breakup of drops and bubbles
in turbulent suspensions. The model is based on the theory of isotropic turbulence and
probability. The breakup kernel is modeled as:
(Eqn. 150)
where:
(Eqn. 151)
is the dimensionless size of eddies in the inertial subrange of isotropic turbulence. The
lower limit of the integration is given by:
(Eqn. 152)
where
(Eqn. 153)
14 1754 1350 454
15 1796 1450 572
16 1837 1550 721
17 1875 1650 908
18 1912 1750 1144
19 1948 1850 1442
20 1983 1950 1817
Group Equal Mass Equal Diameter Geometric Mass
f BVm jmi------=
g vi f BV vi;( ) 0.923FB 1 rd–( )εc
di2
-----⎝ ⎠⎜ ⎟⎛ ⎞ 1 3⁄ 1 ξ+( )2
ξ11 3⁄-------------------e χ– ξdξmin
1∫=
χ12 f BV
2 3⁄ 1 f BV–( )2 3⁄ 1–+( )σ
βρcεc2 3⁄ di
5 3⁄ ξ11 3⁄------------------------------------------------------------------------=
ξ
ξmin 11.4 ηdi----=
η 1εc----νc
3⎝ ⎠⎛ ⎞ 1 4⁄
=
Multiphase Flow Theory: Multiple Size Group (MUSIG) Model
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In addition, is a calibration coefficient, , is the continuous-phase eddy
dissipation rate, is the continuous-phase kinematic viscosity, and is the surface
tension coefficient.
User DefinedModels
A custom model for the breakup kernel may also be provided. The model may be
a CEL expression or User Routine and may be a function of the diameter and/or mass of
groups and as well as any fluid variable.
Coalescence Models
ANSYS CFX-Solver supports 2 coalescence models.
Prince andBlanch Model
The model of Prince and Blanch [62] assumes that the coalescence of two bubbles occurs in
three steps. First, the bubbles collide trapping a small amount of liquid between them. This
liquid film then drains until the liquid film separating the bubbles reaches a critical
thickness. The film them ruptures and the bubbles join together.
The coalescence kernel is therefore modeled by a collision rate of two bubbles and a
collision efficiency relating to the time required for coalescence:
(Eqn. 154)
The collision efficiency is modeled by comparing the time required for coalescence with
the actual contact time during the collision :
(Eqn. 155)
(Eqn. 156)
(Eqn. 157)
where is the initial film thickness, is the critical film thickness when rupture occurs,
and is the equivalent radius:
(Eqn. 158)
FB β 2= εc
νc σ
g vi v j;( )
i j
Q vi v j;( ) θijT θij
B θijS
+ +( )ηij=
tij
τij
ηij etij τij⁄–
=
tij
ρcrij3
16σ----------
⎝ ⎠⎜ ⎟⎛ ⎞
1 2⁄
lnh0h f-----⎝ ⎠
⎛ ⎞=
τijrij
2 3⁄
εc1 3⁄------------=
h0 h f
rij
rij12-- 1
ri--- 1
r j----+⎝ ⎠
⎛ ⎞⎝ ⎠⎛ ⎞ 1–
=
Multiphase Flow Theory: The Algebraic Slip Model
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The turbulent contributions to collision frequency are modeled as:
(Eqn. 159)
where the cross-sectional area of the colliding particles is defined by:
(Eqn. 160)
the turbulent velocity is given by:
(Eqn. 161)
and is a calibration factor. The buoyancy contribution to collision frequency is
modeled as:
(Eqn. 162)
where:
(Eqn. 163)
and is a calibration factor.
The shear contribution to collision frequency is currently neglected.
User-definedModels
A custom model for the coalescence rate kernel may also be provided. The model
may be a CEL expression or User Routine involving the diameter and/or mass of groups
and as well as any fluid variable. Note that the model must give symmetric coalescence
rates ( ).
The Algebraic Slip Model
Models for algebraic slip were first introduced by Ishii [59] Manninen and Taivassalo [60],
provide a more general formulation which forms the basis for the implementation in ANSYS
CFX.
θijT FCT Sij uti
2 utj2
+( )1 2⁄
=
Sijπ4--- di d j+( )2
=
uti 2εc1 3⁄ di
1 3⁄=
FCT
θijB FCBSij U rj U ri–=
U ri2.14σρcdi
------------- 0.505g di+=
FCB
Q vi v j;( )
i
j
Q vi v j;( ) Q v j vi;( )=
Multiphase Flow Theory: The Algebraic Slip Model
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Phasic Equations
A starting point is to review the equations for multiphase flow. The continuity equation for
phase is:
(Eqn. 164)
and the momentum equation is:
(Eqn. 165)
where represents momentum transfer with other phases.
Bulk Equations
A bulk continuity equation is derived by summing (Eqn. 164) over all phases:
(Eqn. 166)
and a bulk momentum equation by summing (Eqn. 166) over all phases:
(Eqn. 167)
where:
(Eqn. 168)
α
ραrα( )∂t∂
-------------------ραrαuα
i( )∂
xi∂--------------------------+ 0=
ραrαuαi( )∂
t∂--------------------------
ραrαuαj uα
i( )∂
x j∂---------------------------------+ rα
p∂xi∂
-------–rατα
ji( )∂
x j∂------------------- rαρα g i Mα
i+ + +=
Mαi
ρm∂t∂
----------ρmum
i( )∂
xi∂----------------------+ 0=
ρmumi( )∂
t∂----------------------
ρmumj um
i( )∂
x j∂-----------------------------+ p∂
xi∂-------–
τmji τD
ji+( )∂
x j∂-------------------------- ρm gi
+ +=
ρm rαραα∑=
ρmumi rαραuα
i
α∑=
τm rαταα∑=
τDji rαρα uα
i umi
–( )uαj
α∑–=
Multiphase Flow Theory: The Algebraic Slip Model
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Drift and Slip Relations
You now define the slip velocity to be the phasic velocity relative to the continuous phase:
(Eqn. 169)
and the drift velocity as:
(Eqn. 170)
The slip and drift velocities are related by:
(Eqn. 171)
With these relationships, the phasic continuity equation may be written in terms of mass
fraction and drift velocity as:
(Eqn. 172)
Derivation of the Algebraic Slip Equation
The phasic and bulk momentum equations are first transformed to nonconservative form
by combining with the phasic and bulk continuity equations. The phasic momentum
equation then becomes:
(Eqn. 173)
and the bulk momentum equation becomes:
(Eqn. 174)
uSαi uα
i uci
–=
uDαi uα
i umi
–=
uDαi uSα
i Y αuSαi
α∑–=
ρmY α( )∂t∂
----------------------xi∂∂ ρmY α um
i uDαi
+( )( )+ 0=
ραrαuα
i∂t∂
--------- ραrαuαj uα
i∂
x j∂---------+ rα
p∂xi∂
-------–rατα
ji( )∂
x j∂------------------- rαρα g i Mα
i+ + +=
ρmum
i∂t∂
--------- ρmumj um
i∂
x j∂---------+ p∂
xi∂-------–
τmji τD
ji+( )∂
x j∂-------------------------- ρm gi
+ +=
Multiphase Flow Theory: The Algebraic Slip Model
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(Eqn. 173) and (Eqn. 174) are combined to eliminate the pressure gradient term, yielding:
(Eqn. 175)
Several assumptions are now made:
1. The dispersed phase is assumed to instantaneously reach its terminal velocity, so thetransient term on the drift velocity is neglected.
2. The approximation is made that:
(Eqn. 176)
3. The viscous stresses and apparent diffusion stresses are neglected.
With these approximations, (Eqn. 175) simplifies to:
(Eqn. 177)
In addition, it is assumed that the interphase momentum transfer is due only to drag and
that the particles are spherical:
(Eqn. 178)
which leads to the following closed relationship for the slip velocity:
(Eqn. 179)
Note that, for rotating reference frames, the apparent accelerations are automatically
included by taking the derivative of the absolute frame velocity rather than relative frame
velocity on the right-hand-side.
Mαi rα ρα
uDαi∂t∂
------------ ρα ρm–( )um
i∂t∂
---------+⎝ ⎠⎜ ⎟⎛ ⎞
=
rα ραuαj uα
i∂
x j∂--------- ρmum
j umi∂
x j∂---------–
⎝ ⎠⎜ ⎟⎛ ⎞
+
rαταji( )∂
x j∂------------------- rα
τmji τD
ji+( )∂
x j∂--------------------------+–
rα ρα ρm–( )gi–
uαj uα
i∂
x j∂--------- um
j umi∂
x j∂---------≈
Mαi rα ρα ρm–( )
umi∂t∂
--------- umj um
i∂
x j∂--------- g i
–+⎝ ⎠⎜ ⎟⎛ ⎞
=
Mαi 3
4--–
rαρcd p
-----------CD
uSα uSαi
=
uSα uSαi 4
3--–
d pρcCD------------ ρα ρm–( )
umi∂t∂
--------- umj um
i∂
x j∂--------- g i
–+⎝ ⎠⎜ ⎟⎛ ⎞
=
Multiphase Flow Theory: Turbulence Modeling in Multiphase Flow
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The effect of in the bulk momentum equation is neglected in the current
implementation.
Turbulence Effects
In Eulerian-Eulerian multiphase, the averaging process used ensures that turbulence
effects do not affect the continuity equation; instead, they appear as an apparent turbulent
dispersion force in the momentum equation.
In the ASM formulation outlined here, turbulent dispersion forces are not considered in the
derivation of slip velocity. Instead, turbulent dispersion is modeled using the same
turbulence model as for multicomponent flows:
(Eqn. 180)
Energy Equation
For multicomponent fluids, the energy equation has a additional term corresponding to
enthalpy transport by species velocities. For standard transported components, this velocity
is modeled using Fick’s law. For ASM components, this term uses the drift velocity.
The following term is therefore added to the right-hand side of (Eqn. 5):
(Eqn. 181)
Wall Deposition
If desired, the dispersed phase which accumulates on a boundary may be removed from the
calculation. A sink for the ASM mass fraction is defined accordingly. The implicit assumption
is that this mass is replaced by an equal mass of the ballast (constraint) component. The
energy equation also removes the enthalpy of the ASM species and adds the enthalpy of the
ballast species.
Turbulence Modeling in Multiphase Flow
This section describes the extension of the single-phase turbulence models to multiphase
simulations. Documentation that describes the theory of single-phase turbulence models,
should be read before continuing with this section. For details, see Turbulence Models
(p. 69).
τDji
ραY α″uαi ″
µtPrt-------
Y α∂
xi∂----------=
x j∂∂ ρmY αuDα
j hαα
N c
∑–
Multiphase Flow Theory: Turbulence Modeling in Multiphase Flow
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Phase-Dependent Turbulence Models
Phase dependent turbulence models can be used in conjunction with the inhomogeneous
model (particle and mixture models) only.
The EddyViscosityHypothesis
The eddy viscosity hypothesis is assumed to hold for each turbulent phase. Diffusion of
momentum in phase is governed by an effective viscosity:
(Eqn. 182)
AlgebraicModels
Zero Equation ModelThe default zero-equation model uses a formula based on geometric length scale and the
mean solution velocity. It is correlated for single-phase turbulent pipe flow. The turbulence
viscosity is modeled as the product of a turbulent velocity scale, , and a turbulence
length scale, , as proposed by Prandtl and Kolmogorov:
(Eqn. 183)
where is a proportionality constant. The velocity scale is calculated to be the maximum
velocity in phase . If you specify a value for the velocity scale, it will be used for all phases.
The length scale is derived using the formula:
(Eqn. 184)
where is the fluid domain volume.
Dispersed Phase Zero Equation Model
(Eqn. 185)
The parameter is a turbulent Prandtl number relating the dispersed phase kinematic
eddy viscosity to the continuous phase kinematic eddy viscosity .
In situations where the particle relaxation time is short compared to turbulence dissipation
time scales, you may safely use the default value . If the particle relaxation time is
long compared to turbulence dissipation time scales, it may be better to use a value of
. This is highly model dependent. Several models are available in the literature.
α
µαeff µα µtα+=
U tα
ltα
µtα ρα f µU tαltα=
f µ
α
ltαV D
1 3⁄
7---------------=
V D
νtdνtcσ------= µtd⇒
ρdρc-----
µtcσ
-------=
σ
νtd νtc
σ 1=
σ 1>
Multiphase Flow Theory: Turbulence Modeling in Multiphase Flow
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Two-EquationModels
For the - model, the turbulent viscosity is modeled as:
(Eqn. 186)
The transport equations for and in a turbulent phase are assumed to take a similar form
to the single-phase transport equations:
(Eqn. 187)
(Eqn. 188)
Definitions of the terms are available. For details, see The k-epsilon model in ANSYS CFX
(p. 75).
The additional terms and represent interphase transfer for and respectively.
These are omitted in ANSYS CFX, though they may be added as user sources.
Other two equation turbulence models are treated in a similar way.
Reynolds StressModels
The multiphase versions of Reynolds stress models are equivalent to the single phase
version, with all flux and volumetric source terms multiplied by volume fractions. Single
phase version information is available. For details, see Reynolds Stress Turbulence Models
(p. 85).
By default, no additional exchange terms are added, though they may be added as user
sources.
Turbulence Enhancement
Sato [22] successfully modeled Particle Induced Turbulence this for bubbly flow using an
enhanced continuous phase eddy viscosity:
(Eqn. 189)
k ε
µtα cµραkα
2
εα--------
⎝ ⎠⎜ ⎟⎛ ⎞
=
k ε
t∂∂ rαραkα( ) rα ραUαkα µ
µtασk--------+⎝ ⎠
⎛ ⎞ kα∇–⎝ ⎠⎛ ⎞
⎝ ⎠⎛ ⎞∇•+
rα Pα ραεα–( ) Tαβk( )
+=
t∂∂ rαραεα( ) rαραUαεα µ
µtασε--------+⎝ ⎠
⎛ ⎞ εα∇–⎝ ⎠⎛ ⎞∇•+
rαεαkα----- Cε1Pα Cε2ραεα–( ) Tαβ
ε( )+=
Tαβk( ) Tαβ
ε( ) k ε
µtc µts µtp+=
Multiphase Flow Theory: Additional Variables in Multiphase Flow
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where is the usual Shear Induced Eddy Viscosity, and is an additional Particle
Induced Eddy Viscosity:
(Eqn. 190)
The variable has a value of 0.6.
Homogeneous Turbulence for Multiphase Flow
In homogeneous multiphase flow, bulk turbulence equations are solved which are the same
as the single phase equations, except that the mixture density and the mixture viscosity are
used. Single phase turbulence model information is available. For details, see Turbulence
Models (p. 69).
For inhomogeneous multiphase flow, it is possible to solve a single turbulence field in a
similar way to homogeneous flow. For details, see Homogeneous Turbulence in
Inhomogeneous Flow (p. 177 in "ANSYS CFX-Solver Modeling Guide").
Additional Variables in Multiphase Flow
When exists in phase , the corresponding field variable is denoted . If it obeys a
transport equation, it is given by:
(Eqn. 191)
Note:
• is the conserved quantity per unit mass of phase .
• is the conserved quantity per unit volume of phase , where
• is the Kinematic Diffusivity for the scalar in phase . This may be set for each
Additional Variable and each phase separately.
• is the external volumetric source term in phase , with units of conserved
quantity per unit volume per unit time.
• represents the total source to due to inter-phase transfer across interfaces
with other phases.
• Although the turbulent Schmidt number, , is shown to apply to phase only, it
currently cannot be specified on a phase-specific basis in ANSYS CFX.
µts µtp
µtp Cµpρcrdd p U d U c–=
Cµp
Φ α Φα
t∂∂ rαραφα( ) rαραUαφα( )∇• rα ραDα
φ( ) µtαSctα----------+⎝ ⎠
⎛ ⎞ φα∇⎝ ⎠⎛ ⎞∇•–+
SαΦ( ) Tα
Φ( )+=
φα α
Φα α Φα ραφα=
DαΦ( ) α
SαΦ( ) α
TαΦ( ) φα
Sctα α
Multiphase Flow Theory: Additional Variables in Multiphase Flow
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As in single phase, Diffusive Transport and Poisson equations are also available for
multiphase additional variables). For details, see Additional Variables (p. 40). Diffusive
transport equations exclude the advection term, turbulent diffusion term, and
contributions due to interphase mass transfer from (Eqn. 191). Poisson equations exclude
these terms as well as the transient term.
Additional Variable Interphase Transfer Models
It is possible for an additional variable to be coupled to a different additional variable
across a phase interface between fluids and .
The total source to per unit volume due to interaction with other phases is given by:
(Eqn. 192)
where:
(Eqn. 193)
The simplest models for interphase transfer between and take the driving force to
be proportional to the difference in bulk additional variable values across the phase
interface:
(Eqn. 194)
(Eqn. 195)
The first of these is used if the additional variable is defined per unit mass. The latter is used
if the additional variable is defined per unit volume.
The coefficients are defined by analogy with heat transfer. For details, see
Inhomogeneous Interphase Heat Transfer Models (p. 148).
Transfer of an additional variable across a phase boundary is described by an additional
variable transfer coefficient . It is the amount of crossing a unit area per unit time
per unit difference in across the phase boundary. Thus:
(Eqn. 196)
(Eqn. 197)
Φα
Ψβ α β
Φα
TαΦ( ) Tαβ
Φ( )
β α≠∑=
TαβΦ( ) T– αβ
Φ( )= Tα
Φ( )
α∑⇒ 0=
Φα Ψβ
TαβΦ( ) cαβ
φ( ) ψβ φα–( )=
TαβΦ( ) cαβ
Φ( ) Ψβ Φα–( )=
cαβΦ( )
ταβ Φα
Φα
TαβΦ( ) ταβAαβ ψβ φα–( ),= for specific variables
TαβΦ( ) ταβAαβ Ψβ Φα–( ),= for volumetric variables
Multiphase Flow Theory: Additional Variables in Multiphase Flow
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So, you have:
(Eqn. 198)
It is often convenient to express the additional variable transfer coefficient in terms of a
dimensionless Sherwood number , analogous to the Nusselt number in heat transfer.
(Eqn. 199)
The diffusivity scale is the kinematic diffusivity for a volumetric variable, and the
dynamic diffusivity for a specific variable.
Particle ModelCorrelations
In the particle model, the diffusivity scale is that of the continuous phase, and the length
scale is the mean diameter of the dispersed phase:
(Eqn. 200)
For laminar forced convection around a spherical particle, theoretical analysis shows that
. For a particle in a moving incompressible Newtonian fluid, the Sherwood number
is a function of the particle Reynolds number and the additional variable Prandtl
number .
Details on the models available in ANSYS CFX for additional variable transfer are available.
For details, see Particle Model Correlations (p. 178 in "ANSYS CFX-Solver Modeling Guide").
Some additional details for the Interface Flux model are provided below.
• Interface Flux
The user specifies directly the interfacial flux from additional variable in fluid 1
to additional variable in fluid 2 of a specified fluid pair. This is the rate of additional
variable transfer per unit time per unit interfacial area from phase 1 to phase 2. Hence,the amount of additional variable transferred to fluid 2 from fluid 1 per unit volume isgiven by:
(Eqn. 201)
may be given as a constant or an expression.
cαβΦ( ) ταβAαβ=
Sh
τ ΓShd
----------=
Γ D
ρD
Γ
d
ταβΓαShαβ
dβ------------------=
Sh 2=
Re
Pr µα ρD( )α⁄=
F12 Φ1
Ψ2
T21 T12– A12F12= =
F12
Multiphase Flow Theory: Additional Variables in Multiphase Flow
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Typically, will be a function of the fluid 1 and fluid 2 additional variable fields, and
possibly other variables. In this case, the user may accelerate convergence of thecoupled solver by also specifying optional fluid 1 and fluid 2 additional variable fluxcoefficients.
(Eqn. 202)
The solver takes the absolute value of these flux coefficients to ensure that they arepositive. This is required for numerical stability. The partial derivatives need not becomputed exactly; it is sufficient for the specified coefficients to simply approximate thepartial derivatives. Specification of additional variable flux coefficients only affects theconvergence rate to the solution of the coupled transfer equations; it does not affect theaccuracy of the converged solution.
For example, the simple model using a transfer coefficient multiplied by bulk specificadditional variable differences may be recovered using:
(Eqn. 203)
Mixture ModelCorrelations
If you are using the mixture model, the Sherwood number is defined in terms of a mixture
diffusivity scale and the mixture length scale:
(Eqn. 204)
Homogeneous Additional Variables in Multiphase Flow
Homogeneous additional variables are assumed to have the same values for all phases, i.e.,
(Eqn. 205)
and are described by the following bulk transport equation:
(Eqn. 206)
where:
(Eqn. 207)
F12
c1∂F12∂Φ1----------- 0≥ ,≈ c2
∂F21∂Ψ2-----------≈ 0,≥ if volumetric
c1∂F12∂φ1----------- 0≥ ,≈ c2
∂F21∂ψ2-----------≈ 0,≥ if specific
F12 F21– c φ1 ψ2–( ),= = c1 c2 c= =
ταβΓαβShαβ
dαβ--------------------- ,= Γαβ rαΓα rβΓβ+=
Φα Φ 1 α N p≤ ≤=
t∂∂ ρφ( ) ρUφ Γ φ∇–( )∇•+ S=
ρ rαραα 1=
N P
∑= U 1ρ--- rαραUα
α 1=
N P
∑= Γ rαΓαα 1=
N P
∑=, ,
Multiphase Flow Theory: Sources in Multiphase Flow
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Homogeneous diffusive transport equations exclude the advection term from (Eqn. 206),
while homogeneous Poisson equations exclude the advection and transient terms.
In most cases, the homogeneous assumption is inappropriate for volumetric additional
variables. For this reason, only specific additional variables are permitted to be
homogeneous in ANSYS CFX.
Sources in Multiphase Flow
The following topics will be discussed:
• Fluid-specific Sources (p. 170)
• Bulk Sources (p. 170)
Fluid-specific Sources
Fluid-specific sources are very similar to those in single phase, except that they are
implemented on a phase basis. For details, see Sources (p. 43). You may need to multiply a
multiphase source by the volume fraction. For details, see Sources in Multiphase Flow
(p. 179 in "ANSYS CFX-Solver Modeling Guide").
Bulk Sources
Sources in multiphase flow often have the property that they scale with volume fraction, i.e.,
as the volume fraction source goes to 0, the source also goes to 0. Bulk sources satisfy this
property. They are applied at a fluid-independent level, and are therefore automatically
added to the equations of all fluids for which they are relevant.
Interphase Mass Transfer
Interphase mass transfer occurs when mass is carried from one phase into another. It is
applicable to both the inhomogeneous and homogeneous multiphase models. For details,
see Interphase Mass Transfer (p. 180 in "ANSYS CFX-Solver Modeling Guide").
Mass transfer is represented by sources in the phasic continuity equations:
(Eqn. 208)
• describes user specified mass sources. For details, see Sources (p. 43).
∂∂t----- rαρα( ) ∇ rαραUα( )•+ Sα Γα+=
Sα
Multiphase Flow Theory: Interphase Mass Transfer
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• is the mass source per unit volume into phase due to interphase mass transfer.
This is expressed as follows:
(Eqn. 209)
• is the mass flow rate per unit volume from phase to phase . You must have:
(Eqn. 210)
As it is important to keep track of the direction of mass transfer processes, it is convenient
to express as follows:
(Eqn. 211)
• The term: represents the positive mass flow rate per unit volume from phase
to phase .
For mass transfer processes across a phase interphase, it is useful to express the volumetric
mass sources in terms of mass fluxes:
(Eqn. 212)
• is the mass flow rate per unit interfacial area from phase to phase , and
is the interfacial area density between the phases.
As interfacial area is commonly proportional to volume fraction, this permits automatic
linearization of mass transfer terms relative to volume fraction.
Secondary Fluxes
The mass source terms affecting the continuity equations, are referred to as primary
mass sources. Clearly, if mass transfer occurs between phases, then this also induces
additional sources and sinks between all other transport equations. These are referred to as
secondary sources, or secondary fluxes, due to mass transfer.
The default form of the secondary source terms for a transported variable is:
(Eqn. 213)
Γα α
Γα Γαββ 1=
N p
∑=
Γαβ β α
Γαβ Γβα–= Γαα 1=
N p
∑ 0=⇒
Γαβ
Γαβ Γαβ+ Γβα
+–=
Γαβ+ 0> β
α
Γαβ mαβAαβ=
mαβ β α Aαβ
Γαβ
φα
SMα Γαβ+ φβ Γβα
+ φα–( )β 1=
N p
∑=
Multiphase Flow Theory: Interphase Mass Transfer
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That is, mass transfer from a phase into a phase carries the bulk conserved quantity
into phase . This is referred to as an upwind formulation, as the upwinded value is carried
out of the outgoing phase, and into the incoming phase.
This default formulation is modified in certain circumstances, in order to take account of
discontinuities in the transported variable at the phase interface, for example, enthalpy in
the case of phase change. For details, see The Thermal Phase Change Model (p. 176).
User Defined Interphase Mass Transfer
For advanced applications, it is possible to specify directly the interphase mass transfer
sources.
If the interphase mass flux between any pair of phases is specified, the volumetric
mass source is computed internally by multiplying by interfacial area density:
(Eqn. 214)
• Alternatively, you may directly specify the volumetric mass source, or interphase mass
flow, .
In both cases, all transport equations are automatically assigned default secondary sources
of the upwinded form:
(Eqn. 215)
This default form of secondary sources may be overridden in CCL.
General Species Mass Transfer
Consider the interphase mass transfer of a component A which is present in two phases
and . There are several different but related variables for measuring the concentration of
component A in a mixture that contains it.
(Eqn. 216)
β α φβ
α
mαβ
Γαβ mαβAαβ=
Γαβ
SMαβ Γαβ+ φβ Γβα
+ φα–( )=
α
β
cAα Molar Concentration of Component A in phase α=
ρAα Mass Concentration of Component A in phase α=
XAα Mole Fraction of Component A in phase α=
Y Aα Mass Fraction of Component A in phase α=
Multiphase Flow Theory: Interphase Mass Transfer
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These are related at follows:
(Eqn. 217)
EquilibriumModels
Consider two phases , , which contain a component A, and consider the situation where
the component A is in dynamic equilibrium between the phases. Typically, the
concentrations of A in each phase are not the same. However, there is a well defined
equilibrium curve relating the two concentrations at equilibrium. This is often, though not
always, expressed in terms of mole fractions:
(Eqn. 218)
For binary mixtures, the equilibrium curve depends on temperature and pressure. For
multicomponent mixtures, it also depends on mixture composition. The equilibrium curve
is in general monotonic and non-linear. Nevertheless, it is convenient to quasi-linearize the
equilibrium relationship as follows:
(Eqn. 219)
Other useful forms of the equilibrium relationship are:
(Eqn. 220)
The various equilibrium ratios are related as follows:
(Eqn. 221)
Raoult’s LawIn gas-liquid systems, equilibrium relationships are most conveniently expressed in terms of
the partial pressure of the solute in the gas phase.
It is well known that, for a pure liquid A in contact with a gas containing its vapor, dynamic
equilibrium occurs when the partial pressure of the vapor A is equal to its saturated vapor
pressure at the same temperature.
ρAα MAcAα=
XAαcAαcα
--------=
Y AαρAαρα
---------=
α β
XAα f XAβ( )=
XAα K Aαβx XAβ= K Aαβ
x mole fraction equilibrium ratio=
Y Aα K Aαβy Y Aβ= K Aαβ
y mass fraction equilibrium ratio=
cAα K Aαβc cAβ= K Aαβ
c molar concentration equilibrium ratio=
ρAα K Aαβρ ρAβ= K Aαβ
ρ mass concentration equilibrium ratio=
K Aαβρ K Aαβ
c cαcβ-----K Aαβ
x ραρβ------K Aαβ
y= = =
Multiphase Flow Theory: Interphase Mass Transfer
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Raoult’s law generalizes this statement for the case of an ideal liquid mixture in contact with
a gas. It states that the partial pressure of solute gas A is equal to the product of its
saturated vapor pressure at the same temperature, and its mole fraction in the liquid
solution .
(Eqn. 222)
If the gas phase is ideal, then Dalton’s law of partial pressures gives:
(Eqn. 223)
and Raoult’s law in terms of a mole fraction equilibrium ratio:
(Eqn. 224)
Henry’s LawIn the case of a gaseous material A dissolved in a non-ideal liquid phase, Raoult’s law needs
to be generalized. Henry’s law states that a linear relationship exists between the mole
fraction of A dissolved in the liquid and the partial pressure of A in the gas phase. This is:
(Eqn. 225)
is Henry’s constant for the component A in the liquid . It has units of pressure, and is
known empirically for a wide range of material pairs, especially for common gases dissolved
in water. It is strongly dependent on temperature.
Henry’s law may also be combined with Dalton’s law in order to express it in terms of a mole
fraction equilibrium ratio:
(Eqn. 226)
Unfortunately, there is no common convention on the definition of Henry’s constant.
Another definition in common use relates the partial pressure in the gas to the molar
concentration in the liquid:
(Eqn. 227)
The two definitions of Henry’s constant are related by:
(Eqn. 228)
PAg
PAsat
XA
PAg PAsatXAl=
PAg XAg Pg=
PAg PAsatXAl=
PAg HxXAl=
Hx l
XAg K xXAl,= K x HAlx
Pg---------=
PAg HccAl=
HxXAl HccAl=
Multiphase Flow Theory: Interphase Mass Transfer
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Two ResistanceModel withNegligible MassTransfer
Due to the presence of discontinuities in concentration at phase equilibrium, it is, in general,
not possible to model multicomponent mass transfer using a single overall mass transfer
coefficient. Instead, it is necessary to consider a generalization of the Two Resistance Model
previously discussed for heat transfer. For details, see The Two Resistance Model (p. 150).
Consider a species A dissolved in two immiscible phases and . The basic assumption is
that there is no resistance to mass transfer at the phase interface, and hence the equilibrium
conditions prevail at the phase interface.
Also, it is assumed in this section that the total mass transfer due to species transfer is
sufficiently small that primary mass sources to the phasic continuity equations are
neglected. Hence, secondary mass fluxes to the species mass transfer equations are also
ignored.
You model the component mass transfer using mass transfer coefficients , , defined
on either side of the phase interface. These are usually defined so that driving forces are
defined in terms of molar concentration differences. Thus, the molar flux of A to phase
from the interface is:
(Eqn. 229)
and the molar flux of A to phase from the interface is:
(Eqn. 230)
Multiplying through by the molar mass of A, this determines the mass fluxes as follows:
Mass flux of A to phase from the interface:
α β
cAβS
cAαS
cAS
cAβ
Phase α Phase βInterface
Molar Concentration of A
kαc kβ
c
α
nAα kαc cAαs cAα–( )=
β
nAβ kβc cAβs cAβ–( )=
α
mAα kαc ρAαs ρAα–( )=
Multiphase Flow Theory: Interphase Mass Transfer
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Mass flux of A to phase from the interface:
(Eqn. 231)
In order to eliminate the unknown interface values, (Eqn. 230) and (Eqn. 231) must be
supplemented by the equilibrium relation between and . This is most conveniently
expressed using the molar concentration equilibrium ratio:
(Eqn. 232)
The mass balance condition:
(Eqn. 233)
may now be combined with the quasi-linearized equilibrium relationship (Eqn. 232) to
determine the interface mass concentrations:
(Eqn. 234)
These may be used to eliminate the interface values in (Eqn. 230) and (Eqn. 231) in order to
express the interfacial mass fluxes in terms of the phasic mass concentrations:
(Eqn. 235)
The Thermal Phase Change Model
This model describes phase change induced by interphase heat transfer in the interior of the
flow. Hence, it may be used to simulate evaporation and condensation, or melting and
solidification. For example, it may be used to model condensation of saturated vapor
bubbles in sub-cooled liquid, or evaporation of saturated bubbles in super-heated liquid.
Modeling advice and consideration information is available. For details, see Thermal Phase
Change Model (p. 181 in "ANSYS CFX-Solver Modeling Guide").
It is essential to consider the heat transfer processes on each side of the phase interface.
Hence, the Two Resistance Model for interphase heat transfer must be used in conjunction
with the Thermal Phase Change model. For details, see The Two Resistance Model (p. 150).
Recall that, in this case, the sensible heat flux to phase from the interface is:
(Eqn. 236)
β
mAβ kβc ρAβs ρAβ–( )=
ραs ρβs
cAαs K ccAβs= ρAαs KcρAβs=⇒
mAα mAβ+ 0=
ρAαs
Kc----------- ρAβs
kαc ρAα kβ
c ρAβ+
K ckαc kβ
c+
-------------------------------------= =
mAα mAβ– kαβc KcρAβ ρAα–( ),= = 1
kαβc
-------- 1
kαc
----- Kc
kβc
------+=
α
qα hα Ts Tα–( )=
Multiphase Flow Theory: Interphase Mass Transfer
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and the sensible heat flux to phase from the interface is:
(Eqn. 237)
• and are the phase and phase heat transfer coefficients respectively.
• The interfacial temperature is determined from considerations of thermodynamic
equilibrium. Ignoring effects of surface tension on pressure, assume , the
saturation temperature.
Note that this is in contrast to the case where there is no mass transfer. In that case, the
interfacial temperature is determined from the sensible heat balance .
In the case of interphase mass transfer, the interphase mass transfer is determined from the
total heat balance, as follows.
Total heat flux to phase from the interface:
(Eqn. 238)
Total heat flux to phase from the interface:
(Eqn. 239)
• denotes mass flux into phase from phase .
• and represent interfacial values of enthalpy carried into and out of the phases
due to phase change, see below for details.
The total heat balance now determines the interphase mass flux:
(Eqn. 240)
The secondary heat flux term must be modified in order to take account of the discontinuity
in static enthalpy due to latent heat between the two phases. This is achieved using a
modification of the upwind formulation (Eqn. 215), due to Prakash [53]. In this formulation,
the bulk fluid enthalpy is carried out of the outgoing phase, as in the default upwind
formulation. However, the saturation enthalpy is carried into the incoming phase. Thus:
(Eqn. 241)
β
qβ hβ Ts Tβ–( )=
hα hβ α β
Ts T sat=
qα qβ+ 0=
α
Qα qα mαβHαs+=
β
Qβ qβ mαβHβs–=
mαβ α β
Hαs Hβs
Qα Qβ+ 0=
mαβqαβ qβα+
Hβs Hαs–------------------------=
mαβ 0> Hαs Hαsat, Hβs Hβ==⇒
mαβ 0< Hαs Hα, Hβs Hβsat==⇒
Multiphase Flow Theory: Interphase Mass Transfer
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This leads to a formulation which is stable both physically and numerically. It implies that
the denominator of (Eqn. 240) is non-zero, being greater than or equal to the latent heat:
.
The Cavitation Model
The tendency for a flow to cavitate is characterized by the cavitation number, defined as:
(Eqn. 242)
where is a reference pressure for the flow (e.g., inlet pressure), is the vapor pressure
for the liquid, and the denominator represents the dynamic pressure. Clearly, the tendency
for a flow to cavitate increases as the cavitation number is decreased.
Cavitation is treated separately from Thermal Phase Change, as the cavitation process is
typically too rapid for the assumption of thermal equilibrium at the interface to be correct.
In the simplest cavitation models, mass transfer is driven by purely mechanical effects,
namely liquid-vapor pressure differences, rather than thermal effects. Current research is
directed towards models which take both effects into account.
In ANSYS CFX, the Rayleigh Plesset model is implemented in the multiphase framework as
an interphase mass transfer model. User defined models can also be implemented.
For cavitating flow, the homogeneous multiphase model is typically used.
The RayleighPlesset Model
The Rayleigh-Plesset equation provides the basis for the rate equation controlling vapor
generation and condensation. The Rayleigh-Plesset equation describing the growth of a gas
bubble in a liquid is given by:
(Eqn. 243)
where represents the bubble radius, is the pressure in the bubble (assumed to be the
vapor pressure at the liquid temperature), is the pressure in the liquid surrounding the
bubble, is the liquid density, and is the surface tension coefficient between the liquid
and vapor. Note that this is derived from a mechanical balance, assuming no thermal
barriers to bubble growth. Neglecting the second order terms (which is appropriate for low
oscillation frequencies) and the surface tension, this equation reduces to:
(Eqn. 244)
L Hβsat Hαsat–=
Cap pv–
12--ρU 2--------------=
p pv
RBd2RB
dt2------------ 3
2--
dRBdt
---------⎝ ⎠⎛ ⎞
2 2σρ f RB------------+ +
pv p–
ρ f--------------=
RB pv
p
ρ f σ
dRBdt
---------23--
pv p–
ρ f--------------=
Multiphase Flow Theory: Interphase Mass Transfer
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The rate of change of bubble volume follows as:
(Eqn. 245)
and the rate of change of bubble mass is:
(Eqn. 246)
If there are bubbles per unit volume, the volume fraction may be expressed as:
(Eqn. 247)
and the total interphase mass transfer rate per unit volume is:
(Eqn. 248)
This expression has been derived assuming bubble growth (vaporization). It can be
generalized to include condensation as follows:
(Eqn. 249)
where is an empirical factor which may differ for condensation and vaporization,
designed to account for the fact that they may occur at different rates (condensation is
usually much slower than vaporization).
Despite the fact that (Eqn. 249) has been generalized for vaporization and condensation, it
requires further modification in the case of vaporization.
Vaporization is initiated at nucleation sites (most commonly non-condensible gases). As the
vapor volume fraction increases, the nucleation site density must decrease accordingly,
since there is less liquid. For vaporization, in (Eqn. 249) is replaced by to
give:
(Eqn. 250)
dV Bdt
----------ddt----- 4
3--πRB
3⎝ ⎠⎛ ⎞ 4πRB
2 23--
pv p–
ρ f--------------= =
dmBdt
----------- ρgdV Bdt
---------- 4πRB2 ρg
23--
pv p–
ρ f--------------= =
NB rg
rg V BNB43--πRB
3 NB= =
m fg NBdmB
dt-----------
3rg ρgRB
---------------- 23--
pv p–
ρ f--------------= =
m fg F3rgρg
RB-------------- 2
3--
pv p–
ρ f------------------- pv p–( )sgn=
F
rg rnuc 1 rg–( )
m fg F3rnuc 1 rg–( )ρg
RB-------------------------------------- 2
3--
pv p–
ρ f------------------- pv p–( )sgn=
Multiphase Flow Theory: Interphase Mass Transfer
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where is the volume fraction of the nucleation sites. (Eqn. 249) is maintained in the
case of condensation. Note that in this model represents the radius of the nucleation
sites.
To obtain an interphase mass transfer rate, further assumptions regarding the bubble
concentration and radius are required. The Rayleigh Plesset cavitation model implemented
in ANSYS CFX uses the following defaults for the model parameters:
•
•
•
•
User DefinedCavitationModels
Additional information on creating a user defined model is available. For details, see User
Defined Cavitation Models (p. 185 in "ANSYS CFX-Solver Modeling Guide").
When using a user defined cavitation model, the ANSYS CFX-Solver will perform generic
linearizations for the volume fraction and volume continuity equations to help stability and
convergence. The saturation pressure is used by the ANSYS CFX-Solver in linearizing the
cavitation rate against pressure.
The Droplet Condensation Model
The Droplet Condensation Model is useful for situations where a dry (or near-saturation)
two-phase flow undergoes rapid pressure reduction leading to nucleation and subsequent
droplet condensation. It is also useful to model additional condensation when droplets are
already present in significant quantities. Typical applications include low-pressure steam
turbines, in which context this model is also referred to as the Nonequilibrium Steam (NES)
model. Such flows are typically transonic. The droplet phase can enter through the inlet or
appear through various nucleation mechanisms, including homogeneous (volumetric) and
heterogeneous (surface) nucleation. In the current release, only homogeneous nucleation
mechanisms have been considered.
The Droplet Condensation Model differs from the Thermal Phase Change model (see The
Thermal Phase Change Model (p. 176)) in that the droplet diameter is calculated as part of
the model rather than as a user input. This leads to improved accuracy. In order to do this a
transport equation for droplet number must also be solved, which has as its source term
nucleation contributions. This model differs from the Equilibrium Phase Change model (see
Equilibrium Phase Change Model (p. 314 in "ANSYS CFX-Solver Modeling Guide")) in that it
does not assume the flow to instantaneously reach equilibrium conditions, and therefore
implicitly includes losses due to thermodynamic irreversibility.
In the following discussion, we consider a homogenous multiphase system, in which the
droplets move with the same velocity as the continuous phase. However, the model has
been generalized to inhomogeneous systems as well.
rnuc
RB
RB 1 µm=
rnuc 5 E 4–=
Fvap 50=
Fcond 0.01=
Multiphase Flow Theory: Interphase Mass Transfer
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The system of equations involves one continuous phase and any number of dispersed
(condensed) phases. The condensed phases travel at the speed of the continuous phase.
Any combination of condensed phases can exist in the solution so that for the continuous
phase, mass conservation becomes:
(Eqn. 251)
where the mass sources are summed over the condensed phases. The condensed
phases can change size by condensation or evaporation.
For a condensed phase, mass conservation is:
(Eqn. 252)
where each dispersed phase has a corresponding number equation of the form:
(Eqn. 253)
and is the nucleation model with units defined as the number of droplets generated per
unit time per unit volume of vapor and is the nucleated droplet mass based on the
critical radius . Note that the droplets are transported with the mixture velocity since no
slip is assumed between the phases. The usual constraint applies for the volume fractions
where:
(Eqn. 254)
Global continuity and momentum equations are also solved as described in Homogeneous
Hydrodynamic Equations (p. 130). The continuous phase energy equation, in total enthalpy
form, is
(Eqn. 255)
∂ρcrc∂t
-------------∂
∂xi------- ρcuirc( )+ Sd m*rcJd+( )
i 1=
nd
∑–=
nd
d
∂ρdrd∂t
--------------∂
∂xi------- ρduird( )+ Sd m*rcJd+=
∂ρcNd∂t
----------------∂
∂xi------- ρcuiNd( )+ ρdrcJd=
Jd
m*
r*
rc rdi 1=
nd
∑+ 1=
∂ρcrcHc∂t
--------------------∂
∂xi------- ρcrcuiHc( )+ rc–( )∂P
∂t------ ∂
∂xi------- Γtrc
∂T∂xi-------⎝ ⎠
⎛ ⎞ SH+ +=
Multiphase Flow Theory: Interphase Mass Transfer
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The Droplet Condensation model can be used for both small and large droplets. However,
small and large droplets use different models for heat transfer and phase change. For large
droplets, the heat transfer and phase change models described by the Thermal Phase
Change model should be used. For small droplets (less than 1 ) the Small Droplet heat
transfer model is appropriate; it sets the droplet temperature to
(Eqn. 256)
where refers to the saturation temperature, refers to the supercooling level in gas
phase, is the droplet radius, and is the critical radius at formation of the dispersed
phase.
For small droplets, the interphase heat and mass transfer models are also modified to
include the influence of the Knudsen ( ) number on the Nusselt number. The
dependence is required since droplet sizes vary significantly from the initial nucleated
radius (in a non-continuum regime) in the range of angstroms. The droplet growth rate is:
(Eqn. 257)
which is subsequently used to compute the interphase mass transfer rate in conjunction
with an interfacial area density to be described later in this section.
The source of droplets into the domain is based on a nucleation model, which for classical
nucleation models has the form of:
(Eqn. 258)
where is a constant determined by the particular nucleation model, is the Gibbs free
energy change at the critical radius conditions, is Boltzmann's constant, and is the
supercooled vapor temperature. To compute the Gibbs free energy change a property
database must be used that evaluates supercooled state properties. This requires an
equation of state for the vapor phase amenable to extrapolation into regions within the
saturation zone. The IAPWS and Redlich Kwong equations of state satisfy this requirement.
µm
Td T s P( ) T scRd
*
Rd------–=
T s T sc
Rd Rd*
Kn Kn
dRddt
---------kc
Rdρd 1 cKn+( )-------------------------------------
Td Tg–
hg hp–------------------⎝ ⎠
⎛ ⎞=
J A ∆G*
kTg----------–
⎝ ⎠⎜ ⎟⎛ ⎞
exp=
A G*
k Tg
Multiphase Flow Theory: Interphase Mass Transfer
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The interaction between the phases by mass transfer depends on calculating the droplet
diameter, which, if a monodispersed distribution is assumed for droplets with a common
origin, can be determined from the droplet number, . The relevant equation is then:
(Eqn. 259)
with an interfacial area density defined as:
(Eqn. 260)
The interfacial mass transfer term can then be computed with the known droplet growth
rate and the interfacial area density:
(Eqn. 261)
which can be used to obtain the heat transfer term:
(Eqn. 262)
where is an upwinded total enthalpy, its value either for the continuous or dispersed
phase depending on the direction of interphase mass transfer. In addition, is the heat
transfer (per unit area) between the dispersed and continuous phase based on:
(Eqn. 263)
Since the droplets in condensing systems are generally quite small (less than 1 ), it is
assumed that the droplet temperature is uniform (a zero resistance model between the
droplet surface and its internal temperature). This implies that almost all of the heat transfer
either comes from the continuous phase during evaporation or goes into it during
condensation.
The Nusselt ( ) number underlying Equation 257 and Equation 263is corrected to
account for droplet sizes that span a wide Knudsen number ( ) range (from
free-molecular to continuum). The Nusselt number applied is:
(Eqn. 264)
Nd
Rd3rd
4πNd--------------⎝ ⎠
⎛ ⎞13--
=
βd3rdRd--------=
Sd ρdβdRddtd
---------=
SH SdHu– βdQd+=
Hu
Qd
Qdkc
Rd 1 cKn+( )------------------------------ Td Tg–( )=
µm
Nu
Kn
Nu 21 cKn+( )
------------------------=
Multiphase Flow Theory: Free Surface Flow
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where is an empirical factor set to 3.18.
Free Surface Flow
This section describes free surface flow theory, which is the most common application of
homogeneous multiphase flow.
Implementation
The implementation of free surface flow in ANSYS CFX is essentially the same as multiphase
(homogeneous or inhomogeneous) along with some special discretization options to keep
the interface sharp. These include:
• A compressive differencing scheme for the advection of volume fractions in the volumefraction equations.
• A compressive transient scheme for the volume fraction equations (if the problem istransient).
• Special treatment of the pressure gradient and gravity terms to ensure that the flow
remain well behaved at the interface.
Surface Tension
The surface tension model used in ANSYS CFX is based on the Continuum Surface Force
model of Brackbill et al [27]. This models the surface tension force as a volume force
concentrated at the interface, rather than a surface force. Consider the free surface interface
shown in the figure below:
Figure 1 Free surface interface
Define a Primary Fluid (the liquid phase) and a Secondary Fluid (usually a gas phase).
The surface tension force given by the Continuum Surface Force model is:
(Eqn. 265)
c
Secondary Fluid, β
Primary Fluid, α
n
θ
α β
Fαβ f αβδαβ=
Multiphase Flow Theory: Free Surface Flow
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where:
(Eqn. 266)
(Eqn. 267)
where is the surface tension coefficient, is the interface normal vector pointing from
the primary fluid to the secondary fluid (calculated from the gradient of a smoothed volume
fraction), is the gradient operator on the interface and is the surface curvature defined
by:
(Eqn. 268)
The two terms summed on the right hand side of (Eqn. 266) reflect the normal and
tangential components of the surface tension force respectively. The normal component
arises from the interface curvature and the tangential component from variations in the
surface tension coefficient (the Marangoni effect).
The term is often called the interface delta function; it is zero away from the interface,
thereby ensuring that the surface tension force is active only near to the interface.
When the interface between the two fluids intersects a wall, it is possible to account for wall
adhesion by specifying the contact angle which the interface makes with the wall through
the primary fluid. The interface normal vector used for the calculations of both curvature
and the surface tension force must satisfy the wall contact angle.
f αβ σαβ καβ nαβ– ∇sσ+=
δαβ ∇rαβ=
σ nαβ
∇S κ
καβ nαβ∇•=
δαβ
Multiphase Flow Theory: Free Surface Flow
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ANSYS CFX-Solver Theory Guide
Particle Transport Theory
Introduction
This chapter describes:
• Lagrangian Tracking Implementation (p. 187)
• Momentum Transfer (p. 189)
• Heat and Mass Transfer (p. 195)
• Basic Erosion Model (p. 205)
• Spray Breakup Models (p. 208)
Particle transport modeling is a type of multiphase model, where particulates are tracked
through the flow in a Lagrangian way, rather than being modeled as an extra Eulerian phase.
The full particulate phase is modeled by just a sample of individual particles. The tracking is
carried out by forming a set of ordinary differential equations in time for each particle,
consisting of equations for position, velocity, temperature, and masses of species. These
equations are then integrated using a simple integration method to calculate the behavior
of the particles as they traverse the flow domain. The following section describes the
methodology used to track the particles
Lagrangian Tracking Implementation
Within the particle transport model, the total flow of the particle phase is modeled by
tracking a small number of particles through the continuum fluid. The particles could be
solid particles, drops or bubbles.
The application of Lagrangian tracking in ANSYS CFX involves the integration of particle
paths through the discretized domain. Individual particles are tracked from their injection
point until they escape the domain or some integration limit criterion is met. Each particle
is injected, in turn, to obtain an average of all particle tracks and to generate source terms
to the fluid mass, momentum and energy equations. Because each particle is tracked from
its injection point to final destination, the tracking procedure is applicable to steady state
flow analysis. The following section describes the methodology used to track the particles.
Particle Transport Theory: Lagrangian Tracking Implementation
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Integration
The particle displacement is calculated using forward Euler integration of the particle
velocity over timestep, .
(Eqn. 1)
where the superscripts o and n refer to old and new values respectively and is the
particle velocity. In forward integration, the particle velocity calculated at the start of the
timestep is assumed to prevail over the entire step. At the end of the timestep, the new
particle velocity is calculated using the analytical solution to Equation 4:
(Eqn. 2)
The fluid properties are taken from the start of the timestep. For the particle momentum, f0
would correspond to the particle velocity at the start of the timestep.
In the calculation of all the forces, many fluid variables, such as density, viscosity and velocity
are needed at the position of the particle. These variables are always obtained accurately by
calculating the element in which the particle is traveling, calculating the computational
position within the element, and using the underlying shape functions of the discretization
algorithm to interpolate from the vertices to the particle position.
Interphase Transfer Through Source Terms
According to Equation 4, the fluid affects the particle motion through the viscous drag and
a difference in velocity between the particle and fluid. Conversely, there is a counteracting
influence of the particle on the fluid flow due to the viscous drag. This effect is termed
coupling between the phases. If the fluid is allowed to influence trajectories but particles do
not affect the fluid, then the interaction is termed one-way coupling. If the particles also
affect the fluid behavior, then the interaction is termed two-way coupling.
The flow prediction of the two phases in one-way coupled systems is relatively
straightforward. The fluid flow field may be calculated irrespective of the particle
trajectories. One-way coupling may be an acceptable approximation in flows with low
dispersed phase loadings where particles have a negligible influence on the fluid flow.
Two-way coupling requires that the particle source terms are included in the momentum
equations. The momentum sources could be due to turbulent dispersion forces or drag. The
particle source terms are generated for each particle as they are tracked through the flow.
Particle sources are applied in the control volume that the particle is in during the timestep.
δt
xin xi
o vpio δt+=
vpi
vp v f vpo v f–( ) δt
τ-----–⎝ ⎠
⎛ ⎞exp τFall 1 δtτ-----–⎝ ⎠
⎛ ⎞exp–⎝ ⎠⎛ ⎞+ +=
Particle Transport Theory: Momentum Transfer
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The particle sources to the momentum equations are obtained by solving transport
equations for the sources. The generic equation for particle sources is:
(Eqn. 3)
Where are the contributions from the particles that are linear in the solution variable
and contains all other contributions. This equation has the same form as the general
particle transport and is solved in the same way as outlined above.
The source to be added to the continuous phase is then S multiplied by the number flow
rate for that particle, which is the mass flow rate divided by the mass of the particle.
In ANSYS CFX, the particle source terms are recalculated each time particles are injected. The
source terms are then retained in memory in order that they may be applied each time the
fluid coefficients are calculated. Thus, the particle sources may be applied even though
particles have not been injected in the current flow calculation.
Momentum Transfer
Consider a discrete particle traveling in a continuous fluid medium. The forces acting on the
particle which affect the particle acceleration are due to the difference in velocity between
the particle and fluid, as well as to the displacement of the fluid by the particle. The equation
of motion for such a particle was derived by Basset, Boussinesq and Oseen for a rotating
reference frame:
(Eqn. 4)
which has the following forces on the right hand side:
• : drag force acting on the particle.
• : buoyancy force due to gravity.
• : forces due to domain rotation (centripetal and Coriolis forces).
• : virtual (or added) mass force. This is the force to accelerate the virtual mass of the
fluid in the volume occupied by the particle. This term is important when the displacedfluid mass exceeds the particle mass, such as in the motion of bubbles.
• : pressure gradient force. This is the force applied on the particle due to the pressure
gradient in the fluid surrounding the particle caused by fluid acceleration. It is onlysignificant when the fluid density is comparable to or greater than the particle density.
• : Basset force or history term which accounts for the deviation in flow pattern from
a steady state. This term is not implemented in ANSYS CFX.
dSPtd
--------- CSφP RS+=
CSφP
RS
mPUPdtd
---------- FD FB FR FVM FP FBA+ + + + +=
FD
FB
FR
FVM
FP
FBA
Particle Transport Theory: Momentum Transfer
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The left hand side of Equation 4 can be modified due to the special form of the virtual mass
term (see Virtual or Added Mass Force (p. 192)) which leads to the following form of the
particle velocity:
(Eqn. 5)
Only a part of the virtual mass term, , remains on the right hand side. The particle and
fluid mass values are given by
(Eqn. 6)
with the particle diameter as well as the fluid and particle densities and . The ratio
of the original particle mass and the effective particle mass (due to the virtual mass term
correction) is stored in
(Eqn. 7)
Using , Equation 5 can be written as
(Eqn. 8)
Each term on the right hand side of Equation 8 can potentially be linearized with respect to
the particle velocity variable , leading to the following equation for each term:
(Eqn. 9)
The following sections show the contribution of all terms to the right hand side values
and the linearization coefficient .
UPdtd
----------1
mPCVM
2-----------mF+
--------------------------------- FD FB F′VM FP+ + +( ) 1mP-------FR+=
F′VM
mPπ6---dP
3 ρP= and mFπ6---dP
3 ρF=
dP ρF ρP
RVMmP
mPCVM
2-----------mF+
---------------------------------ρP
ρPCVM
2-----------ρF+
------------------------------= =
1 RVM–
CVM2
-----------ρF
ρPCVM
2-----------ρF+
------------------------------=
RVM
UPdtd
----------RVMmP
----------- FD FB F′VM FP+ + +( ) 1mP-------FR+=
UP
T R ClinUP+=
R
Clin
Particle Transport Theory: Momentum Transfer
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Drag Force
The aerodynamic drag force on a particle is proportional to the slip velocity, , between
the particle and the fluid velocity:
(Eqn. 10)
where is the drag coefficient and is the effective particle cross section. The drag
coefficient, , is introduced to account for experimental results on the viscous drag of a
solid sphere. The coefficient is calculated in the same way as for Eulerian-Eulerian
multiphase flow.
Using the new variable , this leads to the following contribution to the
right hand side and linearization coefficient of Equation 5:
(Eqn. 11)
(Eqn. 12)
Buoyancy Force
The buoyancy force is the force on a particle immersed in a fluid. The buoyant force is equal
to the weight of the displaced fluid and is given by
(Eqn. 13)
where is the gravity vector.
This leads to the following contribution to the right hand side of Equation 5:
(Eqn. 14)
U s
FD12--CDρFAF U S U S
12--CDρFAF UF UP– UF UP–( )= =
CD AF
CD
D 12--CDρFAF U S=
R D
mPCVM
2-----------mF+
---------------------------------UFRVMmP
-----------DUF= =
ClinD
mPCVM
2-----------mF+
---------------------------------–RVMmP
-----------D–= =
FB mP mF–( )g mP 1ρFρP------–⎝ ⎠
⎛ ⎞ g π6---dP
3 ρP ρF–( )g= = =
g
RρP ρF–
ρPCVM
2-----------ρF+
------------------------------g RVM 1ρFρP------–⎝ ⎠
⎛ ⎞ g= =
Particle Transport Theory: Momentum Transfer
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Rotation Force
In a rotating frame of reference, the rotation term is an intrinsic part of the acceleration in
and is the sum of Coriolis and centripetal forces:
(Eqn. 15)
As described elsewhere (see Pressure Gradient Force (p. 193) and Virtual or Added Mass
Force (p. 192)), the implemented rotation term also contains contributions from the
pressure gradient and the virtual mass force due to the domain rotation which leads to the
following final contribution of the rotation term to the right hand side of Equation 5.
(Eqn. 16)
Using the substitutions:
Equation 16 can be written as:
(Eqn. 17)
where
• if neither pressure gradient nor virtual mass force is taken into account
• if only virtual mass force is taken into account
• if only pressure gradient force is taken into account
• if pressure gradient and virtual mass forces are taken into
account
Virtual or Added Mass Force
This force is caused by the fact that the particle has to accelerate some of the surrounding
fluid, leading to an additional drag which has the following form:
(Eqn. 18)
FR mP 2Ω UP×– Ω Ω rP××–( )=
R 2Ω UP×– Ω Ω rP××–=
1 RVM–( ) 2Ω UF×– Ω Ω rP××–( )– (virtual mass force)
RVMρFρP------– 2Ω UF×– Ω Ω rP××–( ) (pressure gradient term)
RCor,P 2Ω UP×–=
RCor,F 2Ω UF×–=
RCent Ω Ω rP××–=
R RCor,P C 1–( )RCor,F CRCent+ +=
C 1=
C RVM=
C 1 ρF ρP⁄–=
C RVM 1 ρF ρP⁄–( )=
FVMCVM
2-----------mF
UFdtd
----------UPdtd
----------–⎝ ⎠⎛ ⎞=
Particle Transport Theory: Momentum Transfer
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If the virtual mass force is included, the coefficient is normally set to 1. However, when
the virtual mass force is not included, then effectively has the value zero, and is
equal to 1. The second part of the right hand side contains the particle velocity deviation,
hence, it can be brought to the left hand side of the particle momentum equation (see
Equation 5), leading to a modified effective mass. Considering only steady state flows, the
remaining term can be written as
(Eqn. 19)
leading to the following contribution to the right hand side of Equation 5:
(Eqn. 20)
with
(Eqn. 21)
necessary if the particles are solved in a rotating system. The term is shifted to the
rotation term (see Equation 16), therefore, the implemented virtual mass term reduces to
(Eqn. 22)
Pressure Gradient Force
The pressure gradient force results from the local fluid pressure gradient around the particle
and is defined as:
(Eqn. 23)
This force is only important if large fluids pressure gradients exist and if the particle density
is smaller than or similar to the fluid density. Neglecting diffusive and source terms in the
steady state momentum equation, the pressure gradient can be replaced by the velocity
gradient. Assuming a constant fluids density, the pressure gradient force can be written as
(Eqn. 24)
CVM
CVM RVM
F′VMCVM
2-----------mF UF∇UF RF–( )=
R 1 RVM–( ) UF∇UF RF–( )=
RF 2Ω UF×– Ω Ω rP××–=
RF
R 1 RVM–( ) UF∇UF( )=
FPmFρF-------– p∇=
FP mF UF∇UF RF–( ) mPρFρP------ UF∇UF RF–( )= =
Particle Transport Theory: Momentum Transfer
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leading to the following contribution to the right hand side of Equation 5:
(Eqn. 25)
with
(Eqn. 26)
necessary if the particles are solved in a rotating system. The term is shifted to the
rotation term (see Equation 16), therefore, the implemented virtual mass term reduces to
(Eqn. 27)
Turbulence in Particle Tracking
The calculation of the instantaneous fluid velocity, in Equation 4, depends on the flow
regime and the type of particle tracking desired (mean or with turbulent dispersion). In
laminar flows or in flows where mean particle tracking is calculated, is equal to the mean
local fluid velocity, , surrounding the particle. The path of a particle is deterministic (i.e.,
there is a unique path for a particle injected at a given location in the flow).
In turbulent tracking, the instantaneous fluid velocity is decomposed into mean, , and
fluctuating, , components. Now particle trajectories are not deterministic and two
identical particles, injected from a single point, at different times, may follow separate
trajectories due to the random nature of the instantaneous fluid velocity. It is the fluctuating
component of the fluid velocity which causes the dispersion of particles in a turbulent flow.
The model of turbulent dispersion of particles that is used assumes that a particle is always
within a single turbulent eddy. Each eddy has a characteristic fluctuating velocity, ,
lifetime, , and length, le. When a particle enters the eddy, the fluctuating velocity for that
eddy is added to the local mean fluid velocity to obtain the instantaneous fluid velocity used
in Equation 16. The turbulent fluid velocity, , is assumed to prevail as long as the
particle/eddy interaction time is less than the eddy lifetime and the displacement of the
particle relative to the eddy is less than the eddy length. If either of these conditions is
exceeded, the particle is assumed to be entering a new eddy with new characteristic ,
, and le.
R RVMρFρP------ UF∇UF RF–( )=
RF 2Ω UF×– Ω Ω rP××–=
RF
R RVMρFρP------UF∇UF=
v f
v f
v f
v f
v f ′
v f ′
τe
v f ′
v f ′
τe
Particle Transport Theory: Heat and Mass Transfer
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The turbulent velocity, eddy and length and lifetime are calculated based on the local
turbulence properties of the flow:
(Eqn. 28)
(Eqn. 29)
where k and are the local turbulent kinetic energy and dissipation, respectively, and
is a turbulence constant. The factor was chosen to relate the characteristic length
scale to the eddy dissipation length [39]. The variable is a normally distributed random
number which accounts for the randomness of turbulence about a mean value. Because of
this randomness, each component of the fluctuating velocity may have a
different value in each eddy.
Turbulent Dispersion
If turbulent particle dispersion is enabled, you will need to track a much larger number of
particles (usually an order of magnitude higher) since a stochastic method is used. This will
greatly increase computational time; therefore, this is most often performed as a
post-process where there is only one particle iteration.
If turbulent dispersion is used in an iterative situation, it may not be possible to achieve
complete convergence because of the stochastic nature of the sources to the continuous
phase, although the random number generator used in determining the eddies is reset on
each particle iteration.
Heat and Mass Transfer
The following topics will be discussed:
• Simple Mass Transfer (p. 197)
• Heat Transfer (p. 196)
• Liquid Evaporation Model (p. 197)
• Oil Evaporation/Combustion (p. 198)
• Reactions (p. 198)
• Coal Combustion (p. 198)
• Hydrocarbon Fuel Analysis Model (p. 203)
v f ′ Γ 2k 3⁄( )0.5=
leCµ
3 4⁄ k3 2⁄
ε----------------------=
τe le 2k 3⁄( )1 2⁄⁄=
ε Cµ
Cµ3 4⁄
Γ
u′ v′ w′, ,( )
Particle Transport Theory: Heat and Mass Transfer
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Heat Transfer
The rate of change of temperature is governed by three physical processes: convective heat
transfer, latent heat transfer associated with mass transfer, and radiative heat transfer.
The convective heat transfer QC is given by:
(Eqn. 30)
where is the thermal conductivity of the fluid, TG and T are the temperatures of the fluid
and of the particle, and Nu is the Nusselt number given by:
(Eqn. 31)
where CP is the specific heat of the fluid.
The heat transfer associated with mass transfer QM is given by the relation:
(Eqn. 32)
where the sum is taken over all components of the particle for which heat transfer is taking
place. The latent heat of vaporization V is temperature dependent, and is obtained directly
from the MATERIALS information for the liquid in the particle and its vapor.
The radiative heat transfer, , for a particle with diameter Dp, uniform temperature Tp, and
emissivity , is given by:
(Eqn. 33)
where I is the Irradiation Flux on the particle surface at the location of the particle = *
Radiation Intensity, n is the Refractive Index of the fluid, and is the Stefan-Boltzmann
constant. An equivalent amount of heat can be removed from the radiation field.
The rate of change of temperature for the particle is then obtained from:
(Eqn. 34)
where the sum in this equation is taken over all components of the particle including those
not affected by mass transfer.
QC πdλNu TG T–( )=
λ
Nu 2 0.6Re0.5 µCPλ
------⎝ ⎠⎛ ⎞
13--
+=
QMdmC
dt-----------V∑=
QR
εp
Qr14--εpπd p
2 I σnT p4
–( )=
π
σ
mCCP( )∑ tddT QC QM QR+ +=
Particle Transport Theory: Heat and Mass Transfer
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Simple Mass Transfer
Each component of mass being transferred between the continuous and particle phases
satisfies the equation:
(Eqn. 35)
In this equation is the mass of the constituent in the particle, and are the
fractions of mass in the particle and the continuum, is the equilibrium mass fraction ratio,
is the diffusivity of the mass fraction in the continuum, and is the Sherwood number
given by:
(Eqn. 36)
If E is not set by the user, then a value of 1 is used.
The simple model assumes that the mass transfer is driven only by concentration
differences. While this may be appropriate in some situations, such as solids having a small
moisture content, it does not adequately account for the vapor pressure dependence on
particle temperature, which is imported for evaporating liquids. In these situations, the
liquid evaporation model, presented below, is more appropriate.
Liquid Evaporation Model
The liquid evaporation model is a model for particles with heat transfer and one component
of mass transfer, and in which the continuous gas phase is at a higher temperature than the
particles. The model uses two mass transfer correlations depending on whether the droplet
is above or below the boiling point. This is determined through the Antoine equation and is
given by:
(Eqn. 37)
where A, B and C are user-supplied coefficients. The particle is boiling if the vapor pressure,
Pvap, is greater than the gaseous pressure.
When the particle is above the boiling point, the mass transfer is determined by the
convective heat transfer:
(Eqn. 38)
dmCdt
----------- πdρDSh E m⋅ F mFG–( )–=
mC mF mFG
E
ρD Sh
Sh 2 0.6Re0.5 µρD-------⎝ ⎠
⎛ ⎞13--
+=
Pvap Pref A BT C+--------------–⎝ ⎠
⎛ ⎞exp=
tddm QC
V-------–=
Particle Transport Theory: Heat and Mass Transfer
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When the particle is below the boiling point, the mass transfer is given by the formula:
(Eqn. 39)
Here WC and WG are the molecular weights of the vapor and the mixture in the continuous
phase, while X and XG are the molar fractions in the drop and in the gas phase. In either case,
the rate of mass transfer is set to zero when all of the non-base substances in the particle has
evaporated.
Oil Evaporation/Combustion
The oil combustion model uses the particle transport model to track evaporating oil
droplets, which are modeled in a very similar way to the liquid evaporation model, and uses
the eddy dissipation model for the combustion of the volatile gases in the gas phase.
• The Eddy Dissipation Model (p. 229)
• Liquid Evaporation Model (p. 197)
Light OilModification
The light oil modification bases the physical parameters used in the Reynolds number, the
Nusselt number and the Sherwood number on the gas assumed to be in the boundary layer
of the droplet. This, in turn, depends upon the Antoine equation (i.e., if the drop is boiling,
the gas in the boundary layer is all volatiles). In the other extreme, the gas in the boundary
layer consists entirely of the local gas mixture.
Reactions
Arrhenius reactions can be set up in a particle calculation. Reactants must just be in the
particle, but the products can be both in the particle and in the continuous phase. The heat
of the reaction can be shared between the particles and the continuous phase.
Coal Combustion
Coalcombustion -gas phase
Coal combustion is calculated by combining a particle transport calculation of the coal
particles with an eddy dissipation calculation for the combustion of the volatile gases in the
gas phase. Two separate gases are given off by the particles, the volatiles and the char
products which come from the burning of carbon within the particle.
Gas phase combustion is modeled by means of regular single phase reactions. A transport
equation is solved in the fluid for each material given off by the particles. The ‘volatiles’ may
be either a pure substance, a fixed composition mixture, or several independent materials.
Coaldecomposition
Pulverized coal particles are treated in ANSYS CFX as non-interacting spheres with internal
reactions, heat transfer and full coupling of mass, momentum and energy with the gaseous
phase. The combustion of a coal particle is a two stage process: the devolatilization of the
raw coal particle followed by the oxidation of the residual char to leave incombustible ash.
tddm πdDSh
WCWG--------- 1 X–
1 XG–----------------⎝ ⎠
⎛ ⎞log=
Particle Transport Theory: Heat and Mass Transfer
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The devolatilization is usually modeled as a single step or a two step process. The char
oxidation is modeled either as a global reaction, or using an analytical solution for the
diffusion and reaction of oxygen within the pores of the char particle.
The devolatilization and char oxidation processes can occur on time scales of order
milliseconds which are several orders of magnitude less than the typical residence time of
the particle in the furnace. Large variations in time scales can result in numerically stiff
equations, which can cause accuracy problems with explicit integration algorithms. The
ANSYS CFX implementation of the particle transport model bases its timestep on the
reaction rate to ensure that the solution has the required accuracy.
Devolatilization Devolatilization is modeled using the generic Arrhenius reactions capability. Two
alternative devolatilization models are also used for coal combustion. The simpler model is
the single reaction model of Badzioch and Hawksley [78]. At time t, assume that a coal
particle, which originally had unit mass, consists of mass CO of raw coal, mass Cch of residual
char after devolatilization has occurred, and mass A of ash. The rate constant k of the two
reactions determines the rate of conversion of the raw coal:
(Eqn. 40)
the rate of volatiles production is given by:
(Eqn. 41)
and so the rate of char formation is:
(Eqn. 42)
The rate constant kV is expressed in Arrhenius form as:
(Eqn. 43)
where TP is the temperature of coal particle (assumed uniform), and AV and EV are constants,
determined experimentally for the particular coal.
Often the volatiles yield of a particular type of coal is only known from laboratory proximate
analysis, where the heating rate is low and the volatiles escaping from the bed of coal may
undergo secondary reactions including cracking and carbon deposition on solid surfaces. It
has been found experimentally that the potential yield of volatiles from pulverized coal
particles widely dispersed in a gas and heated quickly to typical furnace temperatures can
dCOdt
----------- kV CO–=
dVdt------- Y kV CO=
dCchdt
------------- 1 Y–( )kV CO=
kV AVEVTP------–⎝ ⎠
⎛ ⎞exp=
Particle Transport Theory: Heat and Mass Transfer
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produce a yield greater by as much as a factor of two than the proximate value. If the single
reaction model is used, it is difficult to know which data to use, since the coal particles
experience a wide range of temperatures as they disperse in the furnace.
Bituminous coals generally have a volatile yield, which depends strongly on temperature
and heating rates. In such cases, it is important to take account of this dependence, for
example, by using a multiple reaction model of devolatilization. As an alternative to the
single reaction model, it is better to use the model of Ubhayakar et al. [79] in which two
reactions with different rate parameters and volatile yields compete to pyrolyse the raw
coal. The first reaction dominates at lower particle temperatures and has a lower yield Y1
than the yield Y2 of the second reaction, which dominates at higher temperatures. As a
result, the final yield of volatiles will depend on the temperature history of the particle, and
will increase with temperature, lying somewhere between Y1 and Y2. In this model, the mass
fraction of combustible material (the raw coal) is specified as the mass fraction of volatiles
since all this material could be converted to volatiles.
At time t, assume that a coal particle, which originally had unit mass, consists of mass CO of
raw coal, mass Cch of residual char after devolatilization has occurred, and mass A of ash. The
rate constants k1 and k2 of the two reactions determine the rate of conversion of the raw
coal:
(Eqn. 44)
the rate of volatiles production is given by:
(Eqn. 45)
and so the rate of char formation is:
(Eqn. 46)
The initial value of CO is equal to (1-A). The value of Y1, and parameters A1 and E1 which
define k1 in the Arrhenius equation, analogous to Equation 43, are obtained from proximate
analysis of the coal. Y2, A2 and E2 are obtained from high temperature pyrolysis. Note that
the yields are defined on a dry ash-free (DAF) basis.
There is provision in the code to allow for swelling of a coal particle on devolatilization. The
fractional increase in the mean radius of the coal particles due to devolatilization is required
as a parameter in the input data.
dCOdt
---------- k1 k2+( )– CO=
tddV Y 1k1 Y 2k2+( )CO=
dCchdt
----------- 1 Y 1–( )k1 1 Y 2–( )k2+( )CO=
Particle Transport Theory: Heat and Mass Transfer
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Char oxidation FieldThe rate of diffusion of oxygen is given by kd(Pg-PS), where Pg is the partial pressure of
oxygen in the furnace gases far from the particle boundary layer and PS is the oxygen
pressure at the particle surface. The value of kd is given by:
(Eqn. 47)
where:
• Rp is the particle radius
• TP is the particle temperature
• Tg is the far-field gas temperature
• P is the local pressure
• PA is atmospheric pressure
• Dref is the dynamic diffusivity (recommended value is 1.8e-5 [kg m^-1 s^-1])
• Tref is the reference temperature (recommended value is 293 [K])
• is the exponent with value 0.75
The char oxidation rate per unit area of particle surface is given by kcPS. The chemical rate
coefficient kc is given by:
(Eqn. 48)
where:
• The parameters Ac and Tc depend on the type of coal, and are specified as input
parameters.
• For this model, kd and kc are in units of [kg m^-2 s^-1],
• Recommended values for Ac and Tc are 497 [kg m^-2 s^-1]and 8540K [80]. The overall
char reaction rate of a particle is given by:
(Eqn. 49)
and is controlled by the smaller of the rates kd and kc.
Gibb
The oxidation mechanism of carbon can be characterized by the parameter so that oxides
are produced according to the equation:
(Eqn. 50)
kdDrefRp
----------T p T g+
2Tref-------------------⎝ ⎠
⎛ ⎞ αPAP------=
α
kc AcTCTP-------–⎝ ⎠
⎛ ⎞exp=
kd1– kc
1–+( )
1–molfrcO2
4πRp2 PPA------
φ
φC O2+ 2 φ 1–( )CO 2 φ–( )CO2+→
Particle Transport Theory: Heat and Mass Transfer
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The value of is assumed to depend on the particle temperature TP:
(Eqn. 51)
where the constants are given by Gibb as AS = 2500 and TS = 6240K.
By solving the oxygen diffusion equation analytically, the following equation is obtained for
the rate of decrease in the char mass mc:
(Eqn. 52)
The far field oxygen concentration r• is taken to be the time-averaged value obtained from
the gas phase calculation, and rc is the density of the char. Physically, k1 is the rate of external
diffusion, k2 is the surface reaction rate, and k3 represents the rate of internal diffusion and
surface reaction. These are defined as follows:
(Eqn. 53)
where D is the external diffusion coefficient of oxygen in the surrounding gas. The
coefficient is calculated in the same way as for the Field model, except in this model,
kinematic viscosity is used instead of dynamic viscosity:
(Eqn. 54)
where:
• Dref is the dynamic diffusivity (recommended value is 1.8e-5 [kg m^-1 s^-1])
• Tref is the reference temperature (recommended value is 293 [K])
• is the exponent with value 0.75.
(Eqn. 55)
where kc is the carbon oxidation rate, defined by the modified Arrhenius equation
(Eqn. 56)
φ
2 φ 1–( )2 φ–
-------------------- ASTST p------–⎝ ⎠
⎛ ⎞exp=
td
dmc 3φ1 ε–-----------–
McMO2-----------
ρ∞ρc------ k1
1– k2 k3+( ) 1–+( )
1–mc=
k1D
R2-----=
DDrefρfluid-----------
T p T g+
2Tref-------------------⎝ ⎠
⎛ ⎞=
α
k2 1 ε–( )kcR----=
kc AcT pTcT p------–⎝ ⎠
⎛ ⎞exp=
Particle Transport Theory: Heat and Mass Transfer
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The default values of the model constants are Ac = 14 [m s^-1 K^-1] and Tc = 21580K.
Further:
(Eqn. 57)
where:
(Eqn. 58)
The pore diffusivity, Dp, is computed from external diffusivity, D, according to:
(Eqn. 59)
Note that the units of k1, k2 and k3 for this model are s-1, and the units for kc in equation
Equation 56 differ from those in equation Equation 48 in the Field model.
Radiativepre-heating
The stability of a pulverized coal flame depends on the feedback of heat from the flame zone
to the raw coal as it emerges from the burner. In general, this pre-heating is supplied by a
combination of convective heating by recirculating product gases, as well as by the
absorption of radiation. ANSYS CFX calculates the radiative heating using Equation 33.
The value of the particle emissivity is expected to change as pyrolysis proceeds, i.e., it
varies depending upon the mass fractions of coal and char. The present model assumes a
linear variation in ep from the raw coal value ep(coal) to the value for char ep(char). That is,:
(Eqn. 60)
where fv is the fractional yield of volatiles. Typical values for ep are 1 for coal and 0.6 for char.
Hydrocarbon Fuel Analysis Model
The ‘Hydrocarbon Fuel Analysis’ model allows the user to define all the properties of a solid
or liquid hydrocarbon fuel in a user-friendly way. The solver uses the provided information
to derive the parameters of related objects, e.g., initial particle mass fractions, material
properties of the volatiles released, and stoichiometric or mass coefficients of reactions.
The primary input data corresponds 1-to-1 to what, typically, is available from standard
analysis of the solid or liquid fuel:
• Heating value (higher or lower heating value)
• Proximate analysis (mass fractions of ash, moisture, fixed carbon and volatiles)
• Ultimate analysis (mass fractions of carbon, hydrogen, oxygen, nitrogen, sulphur and
chlorine)
k3 kc β βcoth 1–( ) β2a( )⁄=
β Rkc
Dpεa-------------⎝ ⎠
⎛ ⎞ 0.5=
Dp effic D×=
εp
εp 1 f v–( )εp coal( ) f vεp char( )+=
Particle Transport Theory: Heat and Mass Transfer
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Some additional input data is required by the solver, for which the default values should be
appropriate in many cases:
• Volatiles yield enhancement: Ratio of actual yield under rapid heating to thatdetermined at a slow heating rate in the proximate analysis.
• Average molar mass of volatiles released. Three options are available:
• Automatic: Computes average molar mass from volatiles elementary composition,assuming a mixture of CH4, CO, H2 and H2O
• Value: Use value specified in fuel analysis
• Use Material Definition: Use value defined in volatiles fuel material
• Reference conditions for heating value (temperature and pressure)
• Moisture latent heat in case of higher heating value specified:
• Automatic: Standard value (2.4423 [MJ/kg])
• Value: User value for given temperature and pressure
The above data are used to derive the following quantities:
• Initial mass fractions for particle (ash, char and raw combustible).
Note that initial char mass fraction will typically be zero, as char is produced from rawcombustible during pyrolysis.
• Fuel volatiles material properties:
• Average molar mass
• Specific reference enthalpy (heating value)
• Carbon, hydrogen and oxygen content
• Stoichiometric coefficients for gas phase reactions:
• Fuel volatiles oxidation
• NO reburn by fuel
• Mass coefficients for multiphase reactions:
• Devolatilization (decomposition of raw combustible into char and volatiles)
• Char oxidation
These calculations are performed using a model fuel determined by the fuel analysis data.
The model fuel has all chlorine removed but accounts for the oxygen needed to oxidise
sulphur to SO2. Nitrogen is included into the model fuel if the multiphase reactions are
setup to release HCN to the gas phase, otherwise, the fuel nitrogen is removed.
The total amount of material released to the gas phase during devolatilization is the actual
volatiles yield plus the moisture. Carbon, hydrogen and oxygen content of the volatiles are
computed from ultimate analysis, which in turn defines the stoichiometric coefficients in
the gas phase reactions involving the volatiles material. When the fuel nitrogen model is
enabled, corrections are made in order to account for the carbon and hydrogen released as
HCN.
Particle Transport Theory: Basic Erosion Model
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Basic Erosion Model
The following topics will be discussed:
• Model of Finnie (p. 205)
• Model of Tabakoff and Grant (p. 206)
• Overall Erosion Rate and Erosion Output (p. 208)
Model of Finnie
The wear of a wall due to the erosive effect of particle impacts is a complex function of
particle impact, particle and wall properties. For nearly all metals, erosion is found to vary
with impact angle and velocity according to the relationship [63]:
(Eqn. 61)
where is a dimensionless mass, is the particle impact velocity and is a
dimensionless function of the impact angle. The impact angle is the angle in radians
between the approaching particle track and the wall. The value of the exponent, , is
generally in the range 2.3 to 2.5 for metals.
Finnie’s model of erosive wear [64] relates the rate of wear to the rate of kinetic energy of
impact of particles on the surface, using :
(Eqn. 62)
where:
(Eqn. 63)
Implementationin ANSYS CFX
In ANSYS CFX, the need to adjust the dimension of to obtain a non-dimensional erosion
factor is overcome by specifying:
(Eqn. 64)
where is equal to and defaults to 1 [m/s] in ANSYS CFX.
E kV Pn f γ( )=
E V P f γ( )
n
n 2=
E kV P2 f γ( )=
f γ( ) 13--cos2γ if γtan 1
3-->=
f γ( ) sin(2γ ) 3sin2γ if γtan 13--≤–=
k
EV PV 0-------⎝ ⎠
⎛ ⎞ nf γ( )=
V 01
kn-------⎝ ⎠
⎛ ⎞
Particle Transport Theory: Basic Erosion Model
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Model of Tabakoff and Grant
In the erosion model of Tabakoff and Grant, the erosion rate is determined from the
following relation:
(Eqn. 65)
where:
(Eqn. 66)
(Eqn. 67)
(Eqn. 68)
(Eqn. 69)
Here is the dimensionless mass (mass of eroded wall material divided by the mass of
particle). is the particle impact velocity. is the impact angle in radians between the
approaching particle track and the wall, being the angle of maximum erosion. to ,
and are model constants and depend on the particle/wall material combination.
Implementationin ANSYS CFX
The model constants in the original formulation of Tabakoff and Grant are only valid for
particle velocities specified in feet per second [ft/s]. The Tabakoff formulation is modified in
ANSYS CFX as outlined below:
(Eqn. 70)
E
E k1 f γ( )V P2 cos2γ 1 RT
2–[ ] f V PN( )+=
f γ( ) 1 k2k12 γ π 2⁄γ 0
----------⎝ ⎠⎛ ⎞sin+
2=
RT 1 k4V Psinγ–=
f V PN( ) k3 V P γsin( )4=
k2
1.0 if γ 2γ 0≤
0.0 if γ 2γ 0>⎩⎨⎧
=
E
V P γ
γ 0 k1 k4
k12 γ 0
E f γ( )V PV 1-------⎝ ⎠
⎛ ⎞ 2cos2γ 1 RT
2–[ ] f V PN( )+=
Particle Transport Theory: Basic Erosion Model
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where:
(Eqn. 71)
(Eqn. 72)
(Eqn. 73)
(Eqn. 74)
Note: In ANSYS CFX, the erosive wear is calculated as grams or eroded material per gram ofcolliding particles (not milligrams of eroded material per gram of colliding particles, as maybe found in literature).
Mapping of ANSYS CFX to original Tabakoff constantsTo make the model more general, the model is rewritten so that all model constants have a
dimension of velocity. The following list shows the link between the constants of the
original model and those in ANSYS CFX:
Constants
where:
(Eqn. 75)
(Eqn. 76)
(Eqn. 77)
Value Dimensions ANSYS CFX-Pre Variable
(dimensionless) K12 Constant
(dimensionless)
[Velocity] Reference Velocity 1
[Velocity] Reference Velocity 2
[Velocity] Reference Velocity 3
[deg] Angle of Maximum Erosion
f γ( ) 1 k2k12 γ π 2⁄γ 0
----------⎝ ⎠⎛ ⎞sin+
2=
RT 1V PV 3-------sinγ–=
f V PN( )V PV 2------- γsin⎝ ⎠
⎛ ⎞ 4=
k2
1.0 if γ 2γ 0≤
0.0 if γ 2γ 0>⎩⎨⎧
=
k12
k2
V 1
V 2
V 3
γ 0
V 1 1 k1⁄=
V 2 1 k34( )⁄=
V 3 1 k4⁄=
Particle Transport Theory: Spray Breakup Models
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Overall Erosion Rate and Erosion Output
The erosion of a wall due to a particle is computed from the following relation:
(Eqn. 78)
as an individual representative particle represents many actual particles. Here is the
mass of the particle and is its number rate. The overall erosion of the wall is then the sum
over all particles. This gives an erosion rate in kg/s, and an erosion rate density variable in
the res file and post-processor in kg/s/m2. Note that this erosion rate is only a qualitative
guide to erosion, unless precise values for the model constants are known.
Spray Breakup Models
Spray processes play an important role in many technical systems and industrial
applications. Examples are spray painting or fuel injection systems. The following sections
give an overview about the spray breakup models that are implemented in ANSYS CFX,
Release 11.0.
Primary Breakup/Atomization Models
The main task of atomizer (or primary breakup) models is to determine starting conditions
for the droplets that leave the injection nozzle. These conditions are:
• Initial particle radius
• Initial particle velocity components
• Initial spray angle
These parameters are mainly influenced by the internal nozzle flow (cavitation and
turbulence induced disturbances), as well as by the instabilities on the liquid-gas interface.
A large variety of approaches of different complexities are documented in literature and a
comprehensive model overview is given by Baumgarten et al [120].
In the next section, the following primary breakup models will be described:
• Blob Method (p. 208)
• Enhanced Blob Method (p. 209)
• LISA Model (p. 211)
Blob Method One of the simplest and most popular approaches to define the injection conditions of
droplets is the so-called "blob-method." In this approach, it is assumed that a detailed
description of the atomization and breakup processes within the primary breakup zone of
the spray is not required. Spherical droplets with uniform size, , are injected
that are subject to aerodynamic induced secondary breakup.
ErosionRate E * N * mP=
mP
N
Dp Dnozzle=
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Assuming non-cavitating flow inside the nozzle, it is possible to compute the droplet
injection velocity by conservation of mass:
(Eqn. 79)
is the nozzle cross-section and the mass flow injected through the nozzle.
The spray angle is either known or can be determined from empirical correlations. The
"blob-method" does not require any special settings but is the default injection approach in
ANSYS CFX.
Figure 1 Blob-method
Enhanced BlobMethod
Kuensberg et al [110] have suggested an enhanced version of the blob-method. Similar to
the blob-method, it is assumed that the atomization processes need not be resolved in
detail. Contrary to the standard blob-method this method allows to calculate an effective
injection velocity and an effective injection particle diameter taking into account the
reduction of the nozzle cross section due to cavitation.
During the injection process, the model determines if the flow inside the nozzle is cavitating
or not and dynamically changes the injection particle diameter and the particle injection
velocity. The decision if the flow is cavitating or not is based on the value of the static
pressure at the vena contracta, , that is compared to the vapor pressure, .
(Eqn. 80)
with:
(Eqn. 81)
is the coefficient of contraction that depends on nozzle geometry factor, such as nozzle
length versus nozzle diameter or the nozzle entrance sharpness [114].
U P inital, t( )mnozzle t( )AnozzleρP------------------------=
Anozzle mnozzle
Pvena Pvapor
Pvena t( ) P1ρP2------U vena
2 t( )–=
U vena t( )U mean t( )
Cc----------------------=
Cc
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If is higher than the vapor pressure, , the flow remains in the liquid phase and
the injection velocity is set equal to .
(Eqn. 82)
The initial droplet diameter is equal to the nozzle diameter, .
However, if is lower than , it is assumed that the flow inside the nozzle is
cavitating and the new effective injection velocity, , and injection diameter, , are
computed from a momentum balance from the vena contracta to the nozzle exit (2):
(Eqn. 83)
and:
(Eqn. 84)
with:
(Eqn. 85)
Figure 2 Enhanced blob-method
Input Parameters for the Enhanced Blob MethodThe following information is required for the enhanced blob method:
• Contraction coefficient due to cavitation inside the injection nozzle.
• Injection total pressure of the liquid. This information is required to compute the staticpressure of the liquid at the vena contracta.
• Injection total temperature of the liquid. This information is required to compute thestatic temperature of the liquid at the vena contracta, which is required to determinethe fluid vapor pressure from an Antoine equation (homogeneous binary mixture).
Pvena Pvapor
U mean
U mean t( ) m t( )AnozzleρP-----------------------=
Dnozzle
Pvena Pvapor
U eff Deff
U eff t( )Anozzlem t( )
---------------- Pvapor P2–( ) U vena t( )+=
Deff t( )4Aeff t( )
π--------------------=
Aeff t( ) m U eff t( )ρP( )⁄=
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• Vapor pressure of the particle fluid.
• Normal distance of the pressure probe from the injection center. The fluid pressure at
this position (marked as position 2 in Figure 2) will be used to determine the
acceleration of the liquid from the vena contract to the injection nozzle outlet.
LISA Model The LISA (Linearized Instability Sheet Atomization) model is able to model the major effects
of the kind of primary break-up that is described below and presented in detail by Senecal
et al [126].
In direct injection spark ignition engines, pressure swirl atomizers are often used in order to
establish hollow cone sprays. These sprays are typically characterized by high atomization
efficiencies. With pressure swirl injectors, the fuel is set into a rotational motion and the
resulting centrifugal forces lead to a formation of a thin liquid film along the injector walls,
surrounding an air core at the center of the injector. Outside the injection nozzle, the
tangential motion of the fuel is transformed into a radial component and a liquid sheet is
formed. This sheet is subject to aerodynamic instabilities that cause it to break up into
ligaments.
Figure 3 Pressure Swirl Atomizer
Within the LISA model, the injection process is divided into two stages:
• Film Formation (p. 211)
• Sheet Breakup and Atomization (p. 212)
Both parts of the model are described below.
Film FormationDue to the centrifugal motion of the liquid inside the injector, a liquid film along the injector
walls is formed. The film thickness, , at the injector exit can be expressed by the following
relation:
(Eqn. 86)
h0
m πρuh0 d0 h0–( )=
Particle Transport Theory: Spray Breakup Models
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is the injector exit diameter, and is the mass flow rate through the injector. is the
so-called axial velocity component of the film at the injector exit. This quantity depends on
internal details of the injector and is difficult to calculate. Therefore, in ANSYS CFX, the
approach of Han et al. [127] is used. It is assumed that the total velocity, , is related to the
injector pressure by the following relation:
(Eqn. 87)
is the velocity coefficient, which is computed from:
(Eqn. 88)
Assuming that is known, the total injection velocity can be computed from Equation 86.
The axial film velocity component, , is then derived from
(Eqn. 89)
is the spray angle, which is assumed to be known. At this point, the thickness, , and
axial velocity component of the liquid film are known at the injector exit. The tangential
component of velocity ( ) is assumed to be equal to the radial velocity
component of the liquid sheet downstream of the nozzle exit. The axial component of
velocity is assumed to remain constant.
Sheet Breakup and AtomizationAfter the liquid film has left the injector nozzle, it is subject to aerodynamic instabilities that
cause it to break-up into ligaments. The theoretical development of the model is given in
detail by Senecal et al. [126] and is only briefly repeated here.
The model assumes that a two-dimensional, viscous, incompressible liquid sheet of
thickness moves with velocity through a quiescent, inviscid, incompressible gas
medium. A spectrum of infinitesimal disturbances is imposed on the initially steady motion:
(Eqn. 90)
where is the initial wave amplitude, is the wave number and
is the complex growth rate. The most unstable disturbance, , has the
largest value of and is assumed to be responsible for sheet break-up.
d0 m u
U
U kv2 p∆ρl
----------=
kv
kv max 0.7 4m
d02ρl θcos
----------------------ρl
2 p∆----------,=
p∆
u
u U Θcos=
Θ h0
w U Θsin=
2h U
η η0eikx ωt+=
η0 k 2π( ) λ⁄=
ω ωr iωi+= Ω
ωr
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As derived by Senecal et al. [126], can be computed by finding the maximum of the
following equation:
(Eqn. 91)
with . Once is known, the break-up length, , and the break-up time, , are
given by:
(Eqn. 92)
(Eqn. 93)
is an empirical sheet constant with a default value of 12.
The unknown diameter, , of the ligaments at the break-up point is obtained from a mass
balance. For wavelengths that are long compared to the sheet thickness ( ),
is given by:
(Eqn. 94)
is the wave number corresponding to the maximum growth rate, .
For wavelengths that are short compared to the sheet thickness, is given by:
(Eqn. 95)
In this case, is given by:
(Eqn. 96)
Ω
ωr2vlk
2 kh( )tanhkh( )tanh Q+
------------------------------------–=
4vl2k4 kh( )tanh( )2 Q2U 2k2
– kh( )tanh Q+( ) σk3
ρl--------- QU 2k2
–⎝ ⎠⎜ ⎟⎛ ⎞
–
kh( )tanh Q+-------------------------------------------------------------------------------------------------------------------------------------------------------------+
Q ρg ρl⁄= Ω L τ
L UΩ----
ηbη0------⎝ ⎠
⎛ ⎞ln=
τ 1Ω----
ηbη0------⎝ ⎠
⎛ ⎞ln=
ηb η0⁄( )ln
dl
W eg 27 16⁄<
dl
dl8hK s------=
Ks Ω
dl
dl16hK s---------=
Ks
KsρgU 2
2σ-------------=
Particle Transport Theory: Spray Breakup Models
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The most probable droplet diameter that is formed from the ligaments is determined from
(Eqn. 97)
is the Ohnesorge number that is defined as:
(Eqn. 98)
User Input DataFor the simulation of primary breakup using the LISA model, the following input data is
needed:
• Injection mass flow
• Injection pressure difference
• Nozzle geometry (outer nozzle diameter, injection cone angle)
• Material properties
Specification of the Material Vapor PressureTwo options are available to compute the material vapor pressure: Automatic and Vapor
Pressure. For the optionAutomatic, the particle material vapor pressure is computed from
a homogeneous binary mixture. It is also possible to specify the material vapor pressure
directly by using the option Vapor Pressure.
Primary Breakup Model RestrictionsPrimary break-up models are used to determine starting conditions for the droplets that
leave the injection nozzle. To accomplish this task, the models use averaged nozzle outlet
quantities as input for the calculation of initial droplet diameters, break-up times and
droplet injection velocities. Because all detailed information is replaced by quasi 1D data,
the following restrictions will apply:
• Primary break-up model support will only be available for particle injection regions (PIR)and not for boundary patch injection (BCP).
• On particle injection regions, primary break-up will only be available for cone typeinjection, and only if the nozzle cross sectional area is larger than 0.
• Only single component particle materials are allowed if the Automaticoption is chosen
to compute the particle material vapor pressure inside the nozzle.
Secondary Breakup Models
The following sections give an overview about the currently available secondary breakup
models in ANSYS CFX. Further information about the breakup models can be found in the
provided references.
d0 1.88dl 1 3Oh+( )1 6⁄=
Oh
Oh WeRe
------------=
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BreakupRegimes
The breakup of a liquid jet into droplets is caused by a combination of different
mechanisms: turbulence within the liquid phase, implosion of cavitation bubbles and
external aerodynamic forces acting on the liquid jet. Depending on the injection parameters
such as the relative velocity between liquid and gas, the liquid and gas densities and the
liquid viscosity and surface tension the contribution of each of the above mechanisms to the
spray breakup varies.
Breakup regimes are typically classified in terms of the following dimensionless numbers:
Weber Number:
Ohnesorge Number:
If a droplet is exposed to a gas flow, significant deformation starts at a Weber number of
unity. Above a certain value of the Weber number, the droplet deformation leads to
breakup. Typically, the following breakup regimes are observed [117]:
Numericalapproach tobreakupmodeling
For the numerical simulation of droplet breakup, a so-called statistical breakup approached
is used in ANSYS CFX. In this framework, it is assumed that if a droplet breaks up into child
droplets, the particle diameter is decreased accordingly to the predictions of the used
breakup model. The particle number rate is adjusted so that the total particle mass remains
constant (mass of parent droplet = mass of child droplets). Using this assumption, it is not
required to generate and track new droplets after breakup, but to continue to track a single
representative particle.
Reitz andDiwakarBreakup Model
This model ([115]) distinguishes between two breakup regimes: bag breakup and stripping
breakup. Breakup occurs if a critical particle Weber number has been exceeded.
Independent of the breakup regime, it is assumed that during breakup the following
relation describes the reduction of the particle radius:
(Eqn. 99)
r is the droplet radius prior to breakup, is the new radius for the stable droplet and
is the characteristic breakup time. Values for and t are calculated from the
equations given in the following section:
• Vibrational breakup: < 12
• Bag breakup: 12 < < 50
• Bag-and-stamen breakup: 50 < < 100
• Sheet stripping: 100 < < 350
• Catastrophic breakup: 350 <
WeρFV slip
2 rPσ
-----------------------=
On µρPσDP
---------------------=
WeWeWeWeWe
Σ
td
drP rP rstable–( )–
tbr------------------------------=
rstable
tbr rstable
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Bag breakup
( > ):
(Eqn. 100)
and
(Eqn. 101)
Stripping breakup
( ):
(Eqn. 102)
and
(Eqn. 103)
The model constants , , and of the Reitz & Diwakar breakup model are
accessible via CCL. The standard values of the constants are given in Table 1.
Table 1 Reitz and Diwakar breakup model constants and their default values
SchmehlBreakup Model[112]
In the Schmehl model, the droplet deformation and breakup times are based on
experimental findings of Hsinag et al. [109] and Pilch et al. [117]. It can be shown that
irrespective of the breakup regime, the time to deform a particle from a sphere into a disk
shape is approximately constant:
(Eqn. 104)
Constant Value CCL NameTime Factor for BagBreakup
20 Time Factor forStripping
6.0 Critical Weber Numberfor Bag
0.5 Weber Number Factorfor Stripping
We W ecrit
tbr C1ρPrP
3
2σ-----------=
rstable6σ
ρFV slip2
-----------------=
We Re⁄ Cs1>
tbr C2r
V slip-----------
ρPρP------=
rstableσ2
2ρF2 V slip
3 ν------------------------=
C1 C2 W ecrit Cs1
C1 π
C2
W ecrit
Cs1
ti 1.6*t*=
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with the characteristic time t*
(Eqn. 105)
The second phase of breakup, which is characterized by further distortion of the droplet to
its final destruction is modeled by the following correlations:
(Eqn. 106)
Please note that the last two breakup regimes are an extension to the originally proposed
model by Schmehl and are based on experimental findings given by [117].
For large Ohnesorge numbers (On > 1), the following correlation is used:
(Eqn. 107)
Breakup occurs if with being the time since the last particle breakup. The
droplet size after breakup is computed from the following relation:
(Eqn. 108)
with:
(Eqn. 109)
Child droplets inherit their parent's velocity, plus a velocity component that is given by:
(Eqn. 110)
with ~ 6 at t = and being the droplet diameter before breakup.
t* dPV slip-----------
ρPρF------=
tbrt*------
6 We 12–( ) 0.25– 12 We 18<≤
2.45 We 12–( )0.25 18 We 45<≤
14.1 We 12–( ) 0.25– 45 We 351<≤
0.766 We 12–( )0.25 351 We 2670<≤5.5 2670 We≤⎩
⎪⎪⎪⎨⎪⎪⎪⎧
=
tbrt*------ 4.5 1*1.2On0.74( )=
∆t ti– tbr= ∆t
d32 dP*(1.5On0.2W ecorr0.25– )=
W eCWe
1 1.077On1.6+( )
----------------------------------------=
V Nd p max, d0–
2 tbr ti–( )----------------------------=
d p max, d0 tbr d0
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The velocity component is assumed to be in a plane normal to the parent droplet
velocity direction. Its circumferential orientation within this plane cannot be specified and
is chosen randomly. The total child droplet velocity is then determined from:
(Eqn. 111)
Taylor AnalogyBreakup (TAB)Model [116]
O'Rourke and Amsden proposed the so-called TAB model that is based on the Taylor
analogy. Within the Taylor analogy, it is assumed that the droplet distortion can be
described as a one-dimensional, forced, damped, harmonic oscillation similar to the one of
a spring-mass system. In the TAB model, the droplet deformation is expressed by the
dimensionless deformation , where x describes the deviation of the droplet
equator from its underformed position (see Figure 4). Assuming that the droplet viscosity
acts as a damping force and the surface tension as a restoring force, it is possible to write the
equation of deformation motion as:
(Eqn. 112)
Integration of this equation leads to the following time-dependent particle distortion
equation:
(Eqn. 113)
with:
(Eqn. 114)
(Eqn. 115)
(Eqn. 116)
and are the initial values of distortion and distortion rate of change. For the TAB
model, and are typically taken as zero.
V N
V P new, V P old, V N+=
y 2 x r⁄( )=
y5µp
ρPr2----------- y 8σ
ρPr3----------- y+
2ρg V slip2
3ρg r2---------------------= =
y t( ) W eC e t tD⁄– y0 W eC–( ) ωtcosy0ω-----
y0 W eC–
ωtD-----------------------+⎝ ⎠
⎛ ⎞ ωtsin++=
tD2ρPr2
CdµP--------------=
ω2 Ckσ
ρPr3----------- 1
td2
----–=
W eC WeCf
CkCb-------------=
y0 y0
y0 y0
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During particle tracking, Equation 80 is solved for the dimensionless particle distortion.
Breakup only occurs if the particle distortion y exceeds unity, which means that the
deviation of the particle equator from its equilibrium position has become larger than half
the droplet radius.
The Sauter mean radius of the child droplets after breakup is calculated from the following
expression:
(Eqn. 117)
that is based on the conservation of surface energy and energy bound in the distortion and
oscillation of the parent droplet and surface energy and kinetic energy of the child droplets.
The TAB model has been used to determine the normal velocity of the child droplets after
breakup. At the time of breakup, the equator of the parent droplet moves at a velocity of
in a direction normal to the parent droplet path. This velocity is taken as
the normal velocity component of the child droplets and the spray angle can be
determined from:
(Eqn. 118)
After breakup of the parent droplet, the deformation parameters of the child droplet are set
to .
The following user accessible model constants are available for the TAB breakup model.
rP,ParentrP,Child------------------ 1 0.4K
ρPrP,Parent3
σ------------------------ y0
2 6K 5–120
----------------⎝ ⎠⎛ ⎞+ +=
V N CvCbr y=
θ
θ2---tan
V NV slip-----------=
y 0( ) y 0( ) 0= =
Particle Transport Theory: Spray Breakup Models
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Table 2 TAB breakup model constants and their default values
Figure 4 Particle distortion for the TAB model
ETAB [106] The enhanced TAB model uses the same droplet deformation mechanism as the standard
TAB model, but uses a different relation for the description of the breakup process. It is
assumed that the rate of child droplet generation, , is proportional to the
number of child droplets:
(Eqn. 119)
The constant , depends on the breakup regime and is given as:
(Eqn. 120)
Constant Value CCL Name0.5 Critical Amplitude
Coefficient5.0 Damping Coefficient
1/3 External ForceCoefficient
8.0 Restoring ForceCoefficient
1.0 New Droplet VelocityFactor
10/3 Energy Ratio Factor
Cb
Cd
C f
Ck
Cv
K
dn t( ) dt( )⁄
tdd n t( ) 3Kbrn t( )=
K br
Kbr
k1ω W e W et≤
k2ω We W e W et>⎩⎨⎧
=
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with being the Weber number that divides the bag breakup regime from the stripping
breakup regime. is set to default value of 80. Assuming a uniform droplet size
distribution, the following ratio of child to parent droplet radii can be derived:
(Eqn. 121)
After breakup of the parent droplet, the deformation parameters of the child droplet are set
to . The child droplets inherit a velocity component normal to the path
of the parent droplet with a value:
(Eqn. 122)
where A is a constant that is determined from an energy balance consideration:
(Eqn. 123)
with:
(Eqn. 124)
and being the parent droplet drag coefficient at breakup.
It has been observed that the TAB model often predicts a ratio of child to parent droplet that
is too small. This is mainly caused by the assumption, that the initial deformation parameters
and are zero upon injection, which leads to far too short breakup times. The
largely underestimated breakup times in turn lead to an underprediction of global spray
parameters such as the penetration depth, as well as of local parameters such as the
cross-sectional droplet size distribution. To overcome this limitation Tanner [107] proposed
to set the initial value of the rate of droplet deformation, , to the largest negative root
of Equation 80:
(Eqn. 125)
W et
W et
rP Child,rP Parent,-------------------- e
K brt–=
y 0( ) y 0( ) 0= =
V N Ax=
A2 3 1rP Parent,rP Child,--------------------– 5CDW e 72⁄+
ω2
y2------=
ω2 CkσρPrP Parent,---------------------------=
CD
y 0( ) y 0( )
y 0( )
y 0( ) 1 W eC 1 ωtbucos–( )–[ ] ωωtbusin
-------------------=
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while keeping the initial value of the droplet deformation, . is determined
from the following equation:
(Eqn. 126)
with C = 5.5
The effect of setting to a negative number is to delay the first breakup of the large
initial droplets and to extend their life span, which results in a more accurate simulation of
the jet breakup.
In addition to the TAB model constants, the following user accessible model constants are
available for the ETAB breakup model:
Table 3 ETAB breakup model constants and their default values
CAB A further development of the ETAB model, is the so-called Cascade Atomization and
Breakup Model, CAB. Identical to the ETAB model, the following equation is used to
determine the child droplet size after breakup:
(Eqn. 127)
the main difference being the definition of the breakup constant :
(Eqn. 128)
In addition to the TAB model constants, the following user accessible model constants are
available for the CAB breakup model:
Constant Value CCL Name2/9 ETAB Bag Breakup
Factor2/9 Stripping Breakup
Factor80 Critical Weber Number
for Stripping Breakup
y 0( ) 0= tbu
tbu CρPρF------
dP 0,V P 0,-----------=
y 0( )
K1
K2
Wet
rP Child,rP Parent,-------------------- e
K brt–=
Kbr
Kbr
k1ω 5 W< e 80<
k2ω We 80 W< e 350<
k3ωW e3 4⁄ 350 We<⎩⎪⎪⎨⎪⎪⎧
=
Particle Transport Theory: Spray Breakup Models
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Table 4 CAB breakup model constants and their default values
Dynamic Drag Models
Many particle drag models assume that the droplet remains spherical throughout the
domain. However, this is not always the case and the particle shape may be distorted
significantly. In the extreme case, the particle shape will approach that of a disk. The drag
coefficient is highly dependent on the particle shape and it is therefore desirable to modify
the standard drag laws to account for the effects of droplet distortion.
In ANSYS CFX, the following models are implemented that modify the drag coefficient
depending on the particle distortion:
Liu [108] The drag coefficient is assumed to vary linearly between that of a sphere and that of a disk:
(Eqn. 129)
(Eqn. 130)
y is a measure of the particle distortion. If the droplet is not distorted , then the drag
coefficient of a sphere will be obtained. If the particle is maximally distorted , then
the drag coefficient of a disk will be obtained. This drag model is the standard model for the
TAB, ETAB and CAB breakup models.
Hsiang [109] Based on experimental findings, Hsiang and Faeth suggest to use the following relation to
describe the drag coefficient of the deformed droplet within a Reynolds number range of
1000 to 3300.
(Eqn. 131)
(Eqn. 132)
y is a measure of the particle distortion. Please note that the definition of y has changed in
that denotes the undistorted droplet, while y is equal to 2 if the particle is maximally
distorted. For an undistorted droplet, the drag coefficient of a sphere is obtained. If the
particle is maximally distorted, then the drag coefficient of a disk will be obtained.
Constant Value CCL Name0.05 CAB Bag Breakup Factor
80 Critical Weber Number forStripping Breakup
350 Critical Weber Number forCatastrophic Breakup
K1
Wet
Wet2
Cd droplet, Cd sphere, 1 2.632 y+( ),=
0 y 1< <
y 0=
y 1=
Cd droplet, 0.4 1 2.25 y 1–( )+[ ],=
1 y 2< <
y 1=
Particle Transport Theory: Spray Breakup Models
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Clift [113] For the Reynolds number range between 980 and 10.000 Clift et al. suggest a quadratic
dependency of the drag coefficient on the shape factor, E, that is defined as .
(Eqn. 133)
(Eqn. 134)
E is 1 for a sphere and 0 a disc.
Schmehl [105] The droplet deformation due to external aerodynamic forces leads to a change in the drag
coefficient of the droplet that is assumed to vary between the two limiting geometries:
(Eqn. 135)
with:
(Eqn. 136)
(Eqn. 137)
and:
(Eqn. 138)
Dynamic Drag Law Control
The use of the dynamic drag law is controlled by the following expert parameter:
pt dynamic drag model
This expert parameter can have the listed values:
• 0 Default
• 1 No dynamic drag law
• 2 Liu (default model for TAB, ETAB and CAB)
• 3 Hsiang
• 4 Clift
• 5 Schmehl (default model for Schmehl)
Please note that due the missing information on the droplet deformation, no dynamic drag
law model can be applied to the Reitz & Diwakar breakup model.
E 1 y3⁄=
Cd droplet, 0.445 1 1.63 1 E–( )2+[ ],=
0 E 1< <
Cd droplet, f Cd sphere, 1 f–( )Cd disc,+=
Cd sphere, 0.36 5.48Re 0.573– 24Re------+ +=
Cd disc, 1.1 64πRe----------+=
f E2 1–=
Particle Transport Theory: Spray Breakup Models
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Penetration Depth and Spray Angle
The two most important parameters to globally describe the spray injection are the
penetration depth, S, and the spray angle, . There exists a large number of different
definitions for both quantities; the following are supported by ANSYS CFX:
Note: To use this functionality, a local CCL extension is required that you can obtain fromyour ANSYS representative.
Penetrationdepth alongdirection The penetration of the spray is measured along the user defined spray axis.
Penetrationdepth normal todirection The penetration of the spray is measured normal to the user defined spray axis.
Radialpenetrationdepth The penetration of the spray is measured from a user defined injection point. The spray tip
is located at that position, where a certain, user definable, mass fraction of the spray is
exceeded for the first time.
Figure 5 Penetration depth
θ
SP
SN
SR
Particle Transport Theory: Spray Breakup Models
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Spray Angle The spray half-angle, , is the half-angle of the representative spray cone, that contains a
user definable mass fraction of the spray.
Figure 6 Penetration Angle
Θ
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ANSYS CFX-Solver Theory Guide
Combustion Theory
Introduction
This chapter describes:
• Transport Equations (p. 228)
• Chemical Reaction Rate (p. 228)
• Fluid Time Scale for Extinction Model (p. 229)
• The Eddy Dissipation Model (p. 229)
• The Finite Rate Chemistry Model (p. 230)
• The Combined Eddy Dissipation/Finite Rate Chemistry Model (p. 232)
• Combustion Source Term Linearization (p. 232)
• The Flamelet Model (p. 233)
• Burning Velocity Model (Premixed or Partially Premixed) (p. 239)
• Burning Velocity Model (BVM) (p. 242)
• Laminar Burning Velocity (p. 244)
• Turbulent Burning Velocity (p. 247)
• Spark Ignition Model (p. 250)
• Phasic Combustion (p. 251)
• NO Formation Model (p. 252)
• Chemistry Post-Processing (p. 258)
• Soot Model (p. 259)
This chapter covers the implementation of the combustion models in the ANSYS
CFX-Solver.
This chapter extends the ideas covered in Multicomponent Flow. You should be familiar
with the multicomponent flow chapter before reading this section.
The following sections outline the basis of the implementation of combustion modeling in
ANSYS CFX. First, the transport equations for energy and the components are revisited, then
the chemical reaction rate computation is described, and finally, the computation of the
Combustion Theory: Transport Equations
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rate of progress of a chemical reaction is explained in the context of the Eddy Dissipation
and Finite Rate Chemistry Models. Extinction is modeled by setting the reaction rate locally
to zero.
Transport Equations
Combustion models in ANSYS CFX use the same algorithm used for Multicomponent Fluid
with the addition of a source/sink term due to chemical reactions. The equation of transport
for component I with mass fraction, YI is then:
(Eqn. 1)
where the source term SI is due to the chemical reaction rate involving component I.
Chemical Reaction Rate
In general, chemical reactions can be described in terms of K elementary reactions involving
NC components that can be written as:
(Eqn. 2)
where is the stoichiometric coefficient for component in the elementary reaction .
The rate of production/consumption, , for component can be computed as the sum of
the rate of progress for all the elementary reactions in which component participates:
(Eqn. 3)
where is the elementary reaction rate of progress for reaction , which in ANSYS CFX can
be calculated using the Eddy Dissipation Model or/and the Finite Rate Chemistry Model.
∂ ρY I( )∂t
-----------------∂ ρu jY I( )
∂x j-----------------------+
∂∂x j-------- ΓIeff
∂Y I∂x j---------⎝ ⎠
⎛ ⎞ SI+=
νkI′ I
I A B C …, , ,=
N C
∑ νkI′′ I
I A B C …, , ,=
N C
∑⇔
νkI I k
SI I
I
SI WI νkI′′ νkI
′–( )Rk
k 1=
K
∑=
Rk k
Combustion Theory: Fluid Time Scale for Extinction Model
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Fluid Time Scale for Extinction Model
When the model for flame extinction at high turbulence is activated, local extinction occurs
when . .
While is directly specified, the turbulence time scale is computed from the CFD solution
fields. One possibility is to apply the Kolmogorov time scale:
(Eqn. 4)
An alternative is to use the mixing time scale:
(Eqn. 5)
The Eddy Dissipation Model
The eddy dissipation model is based on the concept that chemical reaction is fast relative to
the transport processes in the flow. When reactants mix at the molecular level, they
instantaneously form products. The model assumes that the reaction rate may be related
directly to the time required to mix reactants at the molecular level. In turbulent flows, this
mixing time is dominated by the eddy properties and, therefore, the rate is proportional to
a mixing time defined by the turbulent kinetic energy, k, and dissipation, .
(Eqn. 6)
This concept of reaction control is applicable in many industrial combustion problems
where reaction rates are fast compared to reactant mixing rates.
In the Eddy Dissipation model, the rate of progress of elementary reaction k, is determined
by the smallest of the two following expressions:
Reactants Limiter
(Eqn. 7)
where [ I ] is the molar concentration of component I and I only includes the reactant
components.
τt τc<
τc
τKolmogorovvε--=
τmixingkε--=
ε
rate εk--∝
Rk Aεk--min I[ ]
vkI′
--------⎝ ⎠⎜ ⎟⎛ ⎞
=
Combustion Theory: The Finite Rate Chemistry Model
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Products Limiter
(Eqn. 8)
where P loops over all product components in the elementary reaction k.
The products limiter is disabled when the model coefficient B is set to a negative value. For
both single step and multi-step reaction schemes, it is turned off by default, (set to -1), but
may be turned on by explicitly setting the model coefficient B to a positive value (although
this is not recommended for multistep reaction schemes).
Maximum Flame Temperature Limiter
Optionally, a maximum flame temperature may be applied for the Eddy Dissipation model.
The reaction rate is smoothly blended to zero when the specified upper temperature limit
is approached. This is implemented by an additional bound added to the minimum
condition in the EDM reaction rate:
(Eqn. 9)
where:
(Eqn. 10)
may be interpreted as a virtual concentration, which vanishes if the temperature is
equal to the maximum flame temperature. is the specific heat capacity of the fluid
mixture at constant pressure and is the reaction heat release per mole.
The Finite Rate Chemistry Model
The Finite Rate Chemistry model, as implemented in ANSYS CFX, assumes that the rate of
progress of elementary reaction k can be reversible only if a backward reaction is defined.
Therefore, the rate of progress Rk, is computed as:
(Eqn. 11)
Rk AB εk--
I[ ]WIP∑
υkI′′ WI
P∑--------------------------
⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞
=
Rk MFT, Aεk--CMFT=
CMFT max Tmax T–( ) 0 K[ ] , ρCP
HR∆-----------⋅=
CMFT
CP
HR∆
Rk Fk I[ ]r ′
kI
I A B …, ,=
N C
∏ Bk I[ ]r ′′
kI
I A B …, ,=
N C
∏–⎝ ⎠⎜ ⎟⎛ ⎞
=
Combustion Theory: The Finite Rate Chemistry Model
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where [ I ] is the molar concentration of component I and Fk and Bk are the forward and
backward rate constants respectively.
r represent the reaction order of component I in the elementary reaction k. This reaction
order is equal to the stoichiometric coefficient for elementary reactions, but it can be
different for certain global reactions.
The only built-in formula for the forward and backward rate constants assumes an Arrhenius
temperature dependence as:
(Eqn. 12)
(Eqn. 13)
where:
• Ak is pre-exponential factor
• is the temperature exponent (dimensionless)
• Ek is the activation energy
• T is the absolute temperature
can also be specified directly without using relations.
Separate sets of coefficients , and are applied to forward and backward rates.
Third Body Terms
In cases that a third body is needed for the reaction to occur, the rate of progress described
earlier is scaled by:
(Eqn. 14)
where is the relative participation of component Ii in reaction k.
For those components that have a high probability to participate in the reaction, the
coefficient is higher than those that rarely participate or do not participate at all
( ). If the third body term is present in a reaction, ANSYS CFX assumes the default
efficiency for all components (Option = Default). Efficiency factors can be specified for
individual species by setting Option to Efficiency Factor List and listing the
components as well as their efficiency factors in the Materials List and Efficiency
Factor List parameters, respectively. If the Efficiency Factor List option is effective,
the default still applies to those components that are not listed.
Fk
Ak
Tβk Ek
RT-------–⎝ ⎠
⎛ ⎞exp=
Bk
Ak
Tβk Ek
RT-------–⎝ ⎠
⎛ ⎞exp=
βκ
Rk
Ak βk Ek
αki I i[ ]i 1=
N
∑
αki
αki
αki 0=
Combustion Theory: The Combined Eddy Dissipation/Finite Rate Chemistry Model
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The Combined Eddy Dissipation/Finite Rate Chemistry Model
The effective reaction rate, for the combined model, is computed to be the minimum of the
Finite Chemistry Rate and the Eddy Dissipation rate.
The Theory documentation for this model is the same as for others:
• The Eddy Dissipation Model (p. 229)
• The Finite Rate Chemistry Model (p. 230)
Combustion Source Term Linearization
The combustion source terms may have a dominant influence on the solution of the scalar
and energy equations. Thus, it is important to treat the combustion source terms carefully
in order to obtain robust convergence of the fluid flow.
A property of multicomponent fluids is that the mass fraction of any given component is
bounded between 0 and 1. Combustion tends to drive reactant concentrations towards the
lower limit and product concentrations toward the upper limit. If the timestep is large, the
combustion sources may cause scalars to exceed these bounds. Thus, the sources may need
to be moderated to maintain physically realistic mass fractions.
The combustion sources in ANSYS CFX have been linearized to prevent the formation of
negative mass fractions. Consider the solution of component I with the source term, R,
which is calculated from (Eqn. 15) in each control volume. To prevent the mass fraction of I
from exceeding the bounds of 0 to 1, the source term is calculated according to:
(Eqn. 15)
where and where
and is a small number (set to 10-6). The combustion reaction rate should approach 0
whenever any of the reactant or product mass fractions approach 0. If the source is positive
(for products), then the first term on the right hand side of (Eqn. 15) is zero and the source is:
(Eqn. 16)
otherwise:
(Eqn. 17)
Thus as , .
sourceRI RI–
2 I ∗--------------------⎝ ⎠
⎛ ⎞ IRI RI+
2 1 I–( )**------------------------⎝ ⎠
⎛ ⎞ 1 I–( )+=
I ∗ max δ I,( )= 1 I–( )** max δ 1, I–( )=
ε
source RI if I 1 δ–≥=
source RI1 I–( )
δ----------------=
products 1→ source 0→
Combustion Theory: The Flamelet Model
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If the source is negative (reactants), then the second term on the right hand side of (Eqn. 15)
is zero and the source is:
(Eqn. 18)
otherwise:
(Eqn. 19)
Thus as , .
This treatment of combustion sources allows larger timesteps to be used in calculating a
steady state solution than would be possible without the linearization.
The Flamelet Model
The Flamelet concept [37] for non premixed combustion describes the interaction of
chemistry with turbulence in the limit of fast reactions (large Damköhler number). The
combustion is assumed to occur in thin sheets with inner structure called Flamelets. The
turbulent flame itself is treated as an ensemble of laminar Flamelets which are embedded
into the flow field.
The Flamelet model is a non equilibrium version of the classical “Burke-Schumann” limit. It
adds new details to the simulation of combustion processes compared to other common
combustion models for the price of the solution of only two scalar equations in the case of
turbulent flow. An arbitrary number of intermediates may be specified as long as their
laminar chemistry is known.
The main advantage of the Flamelet model is that even though detailed information of
molecular transport processes and elementary kinetic reactions are included, the numerical
resolution of small length and time scales is not necessary. This avoids the well-known
problems of solving highly nonlinear kinetics in fluctuating flow fields and makes the
method very robust. Only two scalar equations have to be solved independent of the
number of chemical species involved in the simulation. Information of laminar model flames
are pre-calculated and stored in a library to reduce computational time. On the other hand,
the model is still restricted by assumptions like fast chemistry or the neglecting of different
Lewis numbers of the chemical species.
The coupling of laminar chemistry with the fluctuating turbulent flow field is done by a
statistical method. The PDF used can in principle be calculated at every point in the flow
field by solving a PDF transport equation as shown by Pope and many others. The most
often mentioned advantage of this method is that the non-linear chemical source term
needs no modeling. Even though the method avoids some modeling which is necessary if
using moment closure, it still requires modeling of some of the most important terms, in
particular, the fluctuating pressure gradient term and the molecular diffusion term. If
source RI if I δ≥=
sourceRI I( )
δ--------------=
reactant 0→ source 0→
Combustion Theory: The Flamelet Model
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combustion occurs in thin layers as assumed here, the molecular diffusion term is closely
coupled to the reaction term and the problem of modeling the chemical source term is then
shifted towards modeling the diffusion term.
However, there is no source term in the mixture fraction equation, which is the principal
transport equation in the Flamelet model. Therefore, a presumed beta-PDF, which is a
commonly accepted choice, is used here. Additionally, this avoids the extremely large
computational efforts of calculating the PDF in 3D with a Monte Carlo method.
The following list outlines the assumptions made to derive the Flamelet model:
• Fast Chemistry
• Unity Lewis numbers for all species,
• Combustion is in the Flamelet Regime
• Two feed system, i.e., fluid composition at boundaries must be pure “fuel,” pure“oxidiser” or a linear blend of them.
• Diffusion flames. For premixed or partially premixed combustion, the Flamelet model
can be combined with a model for reaction progress. For details, see Burning Velocity
Model (Premixed or Partially Premixed) (p. 239).
Fluid properties, including temperature and density, are computed from the mean
composition of the fluid in the same way as for other combustion models, such as the Eddy
Dissipation model.
The Flamelet model as implemented in ANSYS CFX can be applied for non-adiabatic
configurations. The only limitation is that changes in the composition of the fluid due to
different temperature and pressure levels are not accounted for. However, the effect of heat
loss and pressure on density and temperature is taken into account. For heat losses
occurring in many combustion devices, the influence of on composition is sufficiently small
to be neglected.
In a large number of industrial combustion devices, pure non-premixed combustion is less
present than premixed or partly premixed combustion. In ANSYS CFX, a model for premixed
and partially premixed combustion is available, which involves the Flamelet model as a
sub-model. For details, see Burning Velocity Model (Premixed or Partially Premixed) (p. 239).
Laminar Flamelet Model for Non Premixed Combustion
A diffusion flame is characterized by the diffusion of reactants into the flame front. While
convective and diffusive time scales are of the same order of magnitude, the chemical time
scales are much smaller for typical combustion processes of interest. Several approaches to
treat chemical reactions have been developed and tested during the last decades.
The assumption of local chemical equilibrium has often been used in modeling the fast
chemistry regime. For hydrocarbon flames, however, the assumption of local chemical
equilibrium results in an over-prediction of intermediates like CO and H2. This suggests that
non equilibrium effects are important in modeling these flames. Further essential non
equilibrium effects are flame extinction, lift-off and blow-out.
Lei 1=
Combustion Theory: The Flamelet Model
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Another well known approach is the flame sheet model of Burke and Schuman often
characterized as `mixed is burned'. Here, only the mixture of the reactants is calculated and
the chemistry is treated as infinite fast and complete when mixing is complete. Therefore,
combustion occurs in an infinitely thin sheet at the surface of stoichiometric mixture. Again
non equilibrium effects are not taken into account, which are important if the strongly
varying time scales of the turbulent flow fields approach those of the chemical reactions.
Linan [42] was the first who incorporated non equilibrium effects in diffusion flames. He
analyzed the inner structure of the thin laminar flame sheet, referred to here as a Flamelet
using an asymptotic description with a large Damköhler number as the expansion
parameter. The Damköhler number is the ratio of flow to chemical time scales:
(Eqn. 20)
Linan’s method is similar to Prandtl's boundary layer theory. The inner layer of the thin
reaction sheet with well defined structure is called “Flamelet” from now on. A more simple
description of flamelets is possible by using the mixture fraction, which is the sum of all
elementary mass fractions:
(Eqn. 21)
which have its origin in a system consisting of fuel inlet (labeled 1) and oxidiser inlet (labeled
2 ). Here is the mass fraction of species i, the mass fraction of a chemical element
(e.g., C or H), the molecular mass, and the number of elements in the molecule
(Eqn. 22)
Assuming equal diffusivities and heat capacities for all chemical components a conservation
equation for the mixture fraction Z can be derived by summing all species conservation
equations, and the chemical source terms therefore cancel exactly. The mixture fraction is
not influenced by chemical reactions because it deals with elements rather than molecules,
and elements are not affected by chemistry.
(Eqn. 23)
Datttc---=
Z jaijM j
Mi-------------Y i
i 1=
n
∑=
Y i Z j j
M aij j i
ZZFuel
ZFuel 1,---------------- 1
ZOxidiserZOxidiser 2,-----------------------–= =
∂ ρZ( )∂t
--------------- ∂ ρZ( )∂x j
---------------+∂
∂x j-------- ρD ∂Z
∂x j--------⎝ ⎠
⎛ ⎞=
Combustion Theory: The Flamelet Model
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The isosurface determines the location of stoichiometric mixture. To be able to
describe the location of the flamelets anywhere in the flow field, a new coordinate system is
introduced here. One of its coordinates is locally perpendicular to the surface of
stoichiometric mixture.
The transformation is shown for the temperature equation as an example. The 3 terms at the
right hand side represent chemical reactions, radiation and the transient pressure gradient,
respectively. The last is important for combustion involving fast changing pressure such as
in closed burning chambers of a piston engine.
(Eqn. 24)
After applying the following transformation rules:
(Eqn. 25)
(Eqn. 26)
(Eqn. 27)
and using the assumption of a constant Lewis number:
(Eqn. 28)
you obtain the temperature equation in the form:
(Eqn. 29)
Z Zst=
ρ∂T∂t------- ρυα
∂T∂xα--------- ∂
∂xα--------- ρD ∂T
∂xα---------⎝ ⎠
⎛ ⎞–+Qkcp------ωk
qRcp-----
1cp-----∂P
∂t------+ +
k 1=
r
∑=
∂∂t----- ∂
∂τ-----
∂Z∂t------- ∂
∂Z-------+=
∂∂xk-------- ∂
∂Zk---------
∂Z∂xk-------- ∂
∂Z------- k 2 3,=( )+=
∂∂x1--------
∂Z∂x1-------- ∂
∂Z-------=
Le λρcpD-------------=
ρ ∂T∂τ------- u2
∂T∂Z2--------- u3
∂T∂Z3---------+ +⎝ ⎠
⎛ ⎞ ∂ ρD( )∂x2
---------------- ∂T∂Z2---------–
∂ ρD( )∂x3
---------------- ∂T∂Z3---------–
ρD ∂Z∂xα---------⎝ ⎠
⎛ ⎞ 2∂2T
∂Z2--------- 2 ∂Z
∂x2-------- ∂2T
∂Z∂Z2---------------- 2 ∂Z
∂x3-------- ∂2T
∂Z∂Z3---------------- ∂2T
∂Z22
--------- ∂2T
∂Z32
---------+ + + +–
1ρcp-------- himi
qRcp-----
1cp-----∂P
∂t------+ +
i 1=
n
∑=
Combustion Theory: The Flamelet Model
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Since the Flamelet is assumed to be thin, only gradients normal to the surface of
stoichiometric mixture are large, and all terms without a second derivative in respect to the
mixture fraction Z can be neglected. When this is done formally by introducing a stretched
coordinate, it turns out that the remaining equation is essentially one dimensional. The
same arguments apply for all other equations as well:
(Eqn. 30)
Non equilibrium effects - the influence of the outer flow field on the inner reaction zone - are
described by the scalar dissipation rate at stoichiometric mixture.
(Eqn. 31)
It represents a reciprocal residence time which is increased by stretch effects of the flow field
and reduced by diffusion. At a critical value of the flame shows a threshold
behavior and extinguishes. The stretch in physical space leads to a reaction zone which is so
thin that the production of heat cannot balance the heat loss caused by conduction. The
temperature drops to unburnt values and the reactions freeze. `Freeze' means they are at
lower temperatures and are much slower than the fluid time scales.
The important conclusion of this derivation is that flamelet structures in the presence of fast
chemistry can be described by one dimensional model flames. This will be used to model
reacting turbulent flow fields.
Coupling of Laminar Flamelet with the Turbulent Flow Field
In the turbulent flow field, the Favre averaged (tilde superscript) mixture fraction equation
is solved:
(Eqn. 32)
The Favre averaging is extensively explained in the theory documentation of ANSYS CFX.
Statistical information on the mixture fraction is obtained from the variance of Z.
(Eqn. 33)
∂T∂t-------
χst2
------∂2T
∂Z2---------–
1ρcp-------- himi
˙ qRcp-----
1cp-----∂P
∂t------+ +
i 1=
n
∑=
χst
χst 2Dst ∇Z( )st2
=
χ χq=
∂ ρZ( )∂t
---------------∂ ρu jZ( )
∂x j--------------------+
∂∂x j-------- µ
µtσZ------+⎝ ⎠
⎛ ⎞ ∂Z∂x j--------
⎩ ⎭⎨ ⎬⎧ ⎫
=
∂ ρZ''2˜
( )∂t
--------------------∂ ρu j
˜ Z''2˜
( )∂x j
-------------------------+∂
∂x j-------- µ
µtσ
Z''2---------+
⎝ ⎠⎜ ⎟⎛ ⎞ ∂Z''
2˜
∂x j-----------
⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫
2µtσZ------ ∂Z
∂x j--------⎝ ⎠
⎛ ⎞ 2ρχ–+=
Combustion Theory: The Flamelet Model
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The structure of this equation is similar to the mixture fraction equation itself except for last
two terms on the right hand side. The first source term is the production and the second
source term models the dissipation of the variance. Here, stands for the scalar dissipation
rate and is modeled in turbulent flow using the empirical relation:
(Eqn. 34)
It includes the effects of strain as well as mixture fraction fluctuations. The standard set of
model coefficients in ANSYS CFX is , , and .
The mean composition of the fluid is computed as a function of mean mixture fraction,
mixture fraction variance and scalar dissipation rat by look-up in a flamelet library:
(Eqn. 35)
The integration over the probability density function (PDF) is not carried out during the
CFD calculation, but is part of the generation process for the Flamelet library. For details, see
Flamelet Libraries (p. 239). The CFD solver looks up the preintegrated values from the
library.
In principle, many types of PDF could be applied, but the most commonly agreed choice is
the Beta-PDF. The shape of is presumed to be that of a beta function ( -function):
(Eqn. 36)
(Eqn. 37)
The Beta-PDF is used for the Flamelet libraries shipped with ANSYS CFX, and for libraries
created with CFX-RIF. For details, see CFX-RIF (p. 275 in "ANSYS CFX-Solver Modeling
Guide").
Note that for the table look-up, the solver is actually using instead of ,i.e., the local
value of the scalar dissipation rate is applied instead of the value at stoichiometric mixture
fraction. This is exact only for stoichiometric mixture or for vanishing variance of mixture
fraction (perfectly premixed case). However, many radicals of interest, for example, OH
radicals, have significant concentrations only at stoichiometry and in its surrounding, which
χ
χ Cχεk--Z''
2=
σZ 0.9= σZ''2
0.9= Cχ 2.0=
Y i˜ Y i
˜ Z Z''2 χst,˜
,( ) Y i Z χst,( ) PZ Z''2
˜,
Z( )∂Z⋅0
1
∫= =
P
P β
PZ Z''2
˜,
Z( ) Za 1– 1 Z–( )b 1–
ψa 1– 1 ψ–( )b 1– ψd0
1∫-----------------------------------------------------=
a Z Z 1 Z–( )
Z''2˜--------------------- 1–
⎝ ⎠⎜ ⎟⎛ ⎞
b, 1 Z–( ) Z 1 Z–( )
Z''2˜--------------------- 1–
⎝ ⎠⎜ ⎟⎛ ⎞
= =
χ χst
Combustion Theory: Burning Velocity Model (Premixed or Partially Premixed)
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makes this approximation acceptable. In principle, could be derived from the solution
fields, but this would introduce errors, too, because it would require either additional
modeling or averaging over the computational domain.
Flamelet Libraries
A flamelet library provides the mean species mass fractions as functions of mean mixture
fraction, variance of mixture fraction and scalar dissipation rate:
(Eqn. 38)
A separate Flamelet library is required for each fuel and each combination of fuel/oxidiser
inlet temperatures and pressure level. The libraries shipped with ANSYS CFX are listed in
Table 1. The files reside in the subdirectory:
<CFXROOT>/etc/reactions-extra/flamelet/
where <CFXROOT> is the location of your ANSYS CFX installation.
Table 1 Flamelet libraries included with ANSYS CFX
If more libraries are needed, these can be created with CFX-RIF. For details, see CFX-RIF
(p. 275 in "ANSYS CFX-Solver Modeling Guide").
Burning Velocity Model (Premixed or Partially Premixed)
The model for premixed or partially premixed combustion can be split into two
independent parts:
• Model for the progress of the global reaction: Burning Velocity Model (BVM), also calledTurbulent Flame Closure (TFC)
• Model for the composition of the reacted and non-reacted fractions of the fluid: Laminar
Flamelet with PDF
The mass fractions in the non-reacted fraction of the fluid, , are obtained by linear
blending of fuel and oxidiser compositions. The species mass fractions in the burned
fraction of the fluid, , are computed by applying the Flamelet model.
Reaction Progress
A single progress variable, , is used to describe the progress of the global reaction:
(Eqn. 39)
χst
Library file name Fuel/Oxidiser Pressure
H2_Air_Tf300K_Tox300K_p1bar.fll hydrogen/air 298 [K] 298 [K] 1 [bar]
CH4_Air_Tf298K_Tox298K_p1bar.fll Methane/air 300 [K] 300 [K] 1 [bar]
Y i˜ Y i
˜ Z Z''2 χst,˜
,( )=
TFuel TOxid
Y i fresh,
Y i burned,
c
Fuel Oxidiser Products→+
Combustion Theory: Burning Velocity Model (Premixed or Partially Premixed)
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The composition of the fluid is determined by blending the compositions of the
non-reacted state (fresh gases) and the reacted state (burned gases), where
corresponds to fresh materials and corresponds to fully reacted materials.
In turbulent flow, a bimodal distribution of is assumed. At any given time and position in
space the fluid is considered to be either fresh materials or fully reacted. This assumption is
justified if the chemical reaction is fast compared to the integral turbulent time scales of the
flow.
Then, the averaged reaction progress variable, , is the probability for the instantaneous
state of the fluid being reacted. The mean species composition of the fluid is computed
according to:
(Eqn. 40)
For example, if , then the fluid at the given position will be fully reacted during 60%
of time and non-reacted and non-reacted during the remaining 40% of time.
The reaction progress variable is computed by solving a transport equation:
(Eqn. 41)
The default value of the turbulent Schmidt number for the reaction progress variable is
.
In the limits of pure fuel and pure oxidizer, the reaction progress is not well defined because
burnt and unburnt conditions correspond to the same physical state:
(Eqn. 42)
(Eqn. 43)
This poses the issue of which boundary value to specify for when the mixture is either
pure fuel or pure oxidizer. Even though different values for correspond to the same
mixture composition, it is still important to impose the proper value on the boundary,
because it controls the combustion regime in the domain after mixing has occurred:
• corresponds with premixed combustion
• corresponds with non-premixed combustion (diffusion flame)
In most cases, is appropriate for fuel inlets (in fact, if for fuel the Flamelet
model could be used and not solve for at all). For oxidizer inlets, the appropriate boundary
value depends on the mixing process in the domain:
c 0=
c 1=
c
c
Y i˜ 1 c–( )Y i fresh, cY i burned,+=
c 0.6=
∂ ρ c( )∂t
--------------∂ ρu j
˜ c( )∂x j
-------------------+∂
∂x j-------- ρD
µtσc-----+⎝ ⎠
⎛ ⎞ ∂ c∂xj------- ωc+=
σc
σc 0.9=
Y i,burned Z 0=( ) Y i,fresh Z 0=( )=
Y i,burned Z 1=( ) Y i,fresh Z 1=( )=
c
c
c 0=
c 1=
c 0= c 1=
c
Combustion Theory: Burning Velocity Model (Premixed or Partially Premixed)
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• for oxidizer premixing with fuel (e.g., primary air)
• for oxidizer mixing with products (e.g., secondary air)
However, which case applies may not be known prior to the simulation. In case of a flow
split, it may even be that both cases occur for the same oxidizer inlet. In this situation, the
artificial distinction between “burnt air” ( ) and “fresh air” ( ) may cause
unphysical behavior such as even fuel being generated by mixing of products with fresh air.
For example, mixing “fresh air” ( , ) with products ( , ) would
result in a mixture that is only partially burnt ( , ). The reaction progress
of the resulting mixture equals the fraction of products , equivalent to fuel being
re-established. This is unphysical. The correct behavior would be obtained by mixing
products with “burnt air” ( , ).
Weighted Reaction Progress
In order to overcome the boundary value issue for reaction progress on oxidizer inlets, the
weighted reaction progress, , is introduced:
(Eqn. 44)
Because for pure oxidizer, the weighted reaction progress is well defined to be
. Linear combination of the transport equations for and yields the following
transport equation for the weighted reaction progress:
(Eqn. 45)
The turbulent Schmidt number for the weighted reaction progress by default is .
Because of the formal derivation of the transport equation, its solution will be equivalent to
solving for reaction progress directly. In other words, introducing the weighted reaction
progress overcomes the boundary value issue for reaction progress without changing the
model. Reaction progress can be restored from mixture fraction and the weighted reaction
progress according to
(Eqn. 46)
The recipe for is ill-posed for . Therefore, numerical treatment is implemented in
the limit of pure oxidiser to smoothly blend both the reaction progress and its gradient to
zero:
c 0=
c 1=
c 1= c 0=
c 0= Z 0= c 1= Z Zst=
c α= Z α Zst⋅=
α 1<
c 1= Z 0=
F
F Z 1 c–( )⋅=
Z 0=
F 0= Z c
∂ ρF( )∂t
---------------∂ ρuj
˜ F( )∂xj
--------------------+∂
∂xj------- ρD
µtσF------+⎝ ⎠
⎛ ⎞ ∂F∂xj------- 2 ρD
µtσF------+⎝ ⎠
⎛ ⎞ ∂Z∂xj------- ∂ c
∂xj-------⋅⎝ ⎠
⎛ ⎞ Zωc–+=
σF 0.9=
c Z F–( ) Z⁄=
c Z 0=
Combustion Theory: Burning Velocity Model (BVM)
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as
Default for ANSYS CFX, Release 11.0 is to solve for weighted reaction progress. Expert
control (CCL) is available for reverting to the standard reaction progress equation:FLUID MODELS:Option = Burning Velocity ModelREACTION PROGRESS VARIABLE:Option = Reaction Progress# default: Weighted Reaction ProgressENDEND
Burning Velocity Model (BVM)
The burning velocity model (BVM), also known as turbulent flame closure (TFC), is used to
close the combustion source term for reaction progress.
(Eqn. 47)
(Eqn. 48)
Where is the density of the unburnt mixture. Note that the molecular diffusion term for
the reaction progress is removed from the transport equation. The diffusive exchange of
species and energy, which makes the flame proceed in space, is already accounted for by
the source term . However, turbulent transport is a convective process and is modeled
using the Eddy Diffusivity approximation.
The model is completed with a closure for the turbulent burning velocity . Accordingly,
this type of model is called Turbulent Burning Velocity Model (BVM). The concept has two
significant advantages compared to models based on molecular reaction rates:
1. In a given configuration, typically varies by only 1 order of magnitude. In contrast,
molecular reaction rates occurring in combustion of hydro-carbonates typically vary inthe computational domain by several orders of magnitude.
2. can be measured directly in experiments, i.e., data is available for the quantity that is
modeled.
Further, the burning velocity directly determines target quantities of a simulation, such as
flame position. Thus, it is easier to derive and fine-tune accurate models for the burning
velocity model than to do so for approaches based on molecular reaction rates.
c 0→ ∂ c∂xj------- 0→, Z 0→
ωc Sc∂
∂xj------- ρD( ) ∂ c
∂xj-------⎝ ⎠
⎛ ⎞–=
Sc ρuST ∇ c=
ρu
Sc
sT
sT
sT
Combustion Theory: Burning Velocity Model (BVM)
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Equivalence Ratio, Stoichiometric Mixture Fraction
Referring to the unburnt mixture, the equivalence ratio describes the ratio of fuel relative
to the amount of fuel that potentially could be burnt with the available oxidizer. For
stoichiometric mixture, the equivalence ratio is defined to be , i.e., the amount of fuel
and oxidizer match such that they could be burnt with neither fuel nor oxidizer left behind.
indicates fuel-rich mixtures (excess fuel), and indicates fuel-lean mixtures
(excess oxidizer). The limits are for pure oxidizer and (infinite) for pure fuel.
When the stoichiometric mixture fraction is known, the local equivalence ratio can be
computed from mixture fraction according to
(Eqn. 49)
The stoichiometric mixture fraction depends on the fuel and the oxygen content in the
oxidizer and is a property of the flamelet library.
• Value
This option allows the user to specify directly the stoichiometric mixture fraction usedfor calculating the equivalence ratio.
• Reactants
For this option, the reactants and their stoichiometric coefficients for a representativeglobal reaction are specified. For example, for a single component fuel,
(Eqn. 50)
or for the generic form,
(Eqn. 51)
The stoichiometric mixture fraction is computed using the reactants stoichiometriccoefficients and the corresponding species mass fractions in the fuel and oxidizerstreams, respectively. The species mass fractions in the fuel and in the oxidizer areobtained from the flamelet library.
• Automatic
This option derives the stoichiometric mixture fraction from the flamelet library
requiring no additional information. The numerical procedure is described below.
φ
φ 1=
φ 1> φ 1<
φ 0= φ ∞=
Zst
Z
φ Z1 Z–------------
1 Zst–
Zst---------------⋅=
Zst
νFFuel νOxO2+ (products)→
ναXαα∑ (products)→
Combustion Theory: Laminar Burning Velocity
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Figure 1 shows the qualitative behavior of mass fractions for fuel and oxygen plotted over
mixture fraction (hydrogen/air in the example).
Figure 1 Oxygen mass fraction over mixture fraction
One can observe that the oxygen concentration is approximately linear below and above
the stoichiometric mixture fraction: linear decay on the lean side and constantly zero on the
fuel side. Obviously, the curvature of the curve is close to zero except near the sharp bend
at stoichiometric mixture fraction. This observation is generalized to establish the following
procedure
The stoichiometric mixture fraction approximated by the point of maximum curvature for
oxygen mass fraction, or , is
(Eqn. 52)
This is a heuristic approach and only provides an approximation. It is recommended to
check for plausibility of the calculated value, which is reported to the solver output file. For
details, see ANSYS CFX Output File (Combustion Runs) (p. 50 in "ANSYS CFX-Solver Manager
User's Guide").
Laminar Burning Velocity
The laminar burning velocity, , is a property of the combustible mixture. It is defined as
the speed of the flame front relative to the fluid on the unburnt side of the flame. The
burning velocity relative to the burnt fluid will be higher by a factor equal to the expansion
ratio, .
Zc
Zst Zc≈∂2Y O2
Zc( )
∂Z2-------------------------, max0 Z 1≤ ≤
∂2Y O2Z( )
∂Z2-----------------------=
sL
sLburnt sL ρu ρb⁄⋅=
Combustion Theory: Laminar Burning Velocity
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Physically, the laminar burning velocity depends on the fuel, the equivalence ratio, the
temperature of the unburnt mixture (preheating) and on pressure. Depending on the
configuration in the simulation, it may be possible sometimes to neglect preheat and
pressure dependencies. However, for partially premixed combustion, it is very important to
account for the dependency on equivalence ratio. Specifically, the flammability limits have
to be obeyed.
Value
User-defined laminar burning velocity, . It is recommended to account for dependency
on fuel/oxidizer ratio by making the expression depend on equivalence ratio or mixture
fraction. Properties of the unburnt mixture (temperature, density, specific heat capacity,
thermal conductivity) may be used in the expression in order to account for preheating or
mixture dependency.
Equivalence Ratio Correlation
The equivalence ratio correlation follows the approach by Metghalchi and Keck [124],
expressing the laminar burning velocity as a base value at reference conditions, ,
multiplied by correction factors for preheat and pressure dependencies:
(Eqn. 53)
The exponents for preheat dependency and for pressure dependency are quadratic
polynomials in equivalence ratio:
(Eqn. 54)
(Eqn. 55)
When all three coefficients are set to zero, then the preheat dependency or the pressure
dependency is disabled. For reference burning velocity, the following options are available:
• Fifth Order Polynomial (p. 246)
• Quadratic Decay (p. 246)
• Beta Function (p. 246)
All three options specify the flammability limits for fuel-lean and for fuel-rich mixtures,
and . The burning velocity is set to zero if the local equivalence ratio is out of
these bounds.
sL
sL,0
sL sL0 Tu
T ref---------⎝ ⎠
⎛ ⎞ α ppref--------⎝ ⎠
⎛ ⎞ β⋅ ⋅=
α a0 a1φ a2φ2+ +=
β b0 b1φ b2φ2+ +=
φflam,l φflam,r
Combustion Theory: Laminar Burning Velocity
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Fifth OrderPolynomial
The laminar burning velocity may be specified as a polynomial up to fifth order:
(Eqn. 56)
This polynomial is evaluated on a specified fit range, . Outside this range,
the burning velocity is modeled to linearly decay to zero at the flammability limit.
Quadratic Decay For quadratic decay the maximum laminar burning velocity at reference conditions, ,
and the corresponding equivalence ratio, , are given. For smaller or larger equivalence
ratio the burning velocity is modeled to decrease according to a quadratic decay coefficient,
:
(Eqn. 57)
This quadratic function is evaluated on a specified fit range, . Outside this
range, the burning velocity is modeled to linearly decay to zero at the flammability limit.
Beta Function The beta function correlation sets the maximum laminar burning velocity, , and the
corresponding equivalence ratio, . A beta function is used to model the velocity decay
to zero at the fuel-lean or fuel-rich flammability limit:
where and .
Metghalchi andKeck
The correlation by Metghalchi and Keck is based on the equivalence ratio correlation with
quadratic decay described above. Predefined sets of coefficients are provided for several
hydrocarbon fuels. The fuel type is characterized by the number of carbon atoms in the fuel
molecule, here called the fuel carbon index. Table 2 lists the coefficients for methane,
propane and iso-octane (gasoline), respectively.
Table 2 Fuel Dependent Coefficients for Metghalchi and Keck Laminar BurningVelocity Correlation
For fuels with other carbon indices the coefficients are obtained by linear interpolation of
the provided values. Fit range, preheat dependency and pressure dependency are modeled
independent of the fuel (Table 3).
sL0 s0φ s1φ s2φ2 s3φ3 s4φ4 s5φ5
+ + + + +=
φfit,l φ φfit,r≤ ≤
smax0
φmax
Cdecay
sL0 s0
max Cdecay φ φmax–( )2–=
φfit,l φ φfit,r≤ ≤
s0max
φmax
sL0 s0
max φ φflam,l–
φmax φflam,l–------------------------------⎝ ⎠
⎛ ⎞ a φflam,r φ–
φflam,r φmax–-------------------------------⎝ ⎠
⎛ ⎞ b⋅ ⋅=
a 2φmax φflam,l–
φflam,r φflam,l–---------------------------------⋅= b 2
φflam,r φmax–
φflam,r φflam,l–---------------------------------⋅=
CarbonIndex
Fuel [m/s] [m/s]
1 Methane 0.35 1.387 1.06 0.533 1.68
3 Propane 0.342 1.387 1.08 0.536 2.50
8 Iso-octane 0.263 0.847 1.13 0.601 3.80
s0max Cdecay φmax φflam,l φflam,r
Combustion Theory: Turbulent Burning Velocity
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Table 3 Common Coefficients for Metghalchi and Keck Laminar Burning VelocityCorrelation
Turbulent Burning Velocity
For turbulent flow, the effective or turbulent burning velocity will differ from the laminar
burning velocity, . Typically turbulence will increase the burning velocity, because
wrinkling of the flame front results in an increased effective flame surface. At very high
turbulence, the opposite effect may occur, leading to a decrease in the effective burning
velocity because of local extinction. A model is required to describe the turbulent burning
velocity as a function of laminar burning velocity and turbulence quantities.
The burning velocity is defined relative to the unburnt fluid. Relative to the burnt fluid, it will
be higher by a factor equal to the fluid expansion ratio, .
Value
This option can be used to implement user models for turbulent burning velocity. Typically,
this will include expressions using laminar burning velocity and turbulence quantities.
Zimont Correlation
The closure developed by Zimont et. al [38] [40] [41] is used for the turbulent burning
velocity:
(Eqn. 58)
The leading factor, , is a modeling coefficient that has the universal value
(default), with the exception of H2/Air flames where is recommended [41].
The stretching factor, , accounts for reduction of the flame velocity due to large strain rate
(large dissipation rate of the turbulent kinetic energy). This effect is modeled in terms of the
probability for turbulence eddy dissipation, , being larger than a critical value . For
, flamelet extinction takes place, while for , the stretching effect is ignored
completely. Assuming a lognormal distribution for , the stretching factor is given by:
(Eqn. 59)
0.7 1.4 2.98 -0.8 0 -0.38 0.22 0
φfit,l φfit,r a0 a1 a2 b0 b1 b2
sT
sL
sTburnt sT ρu ρb⁄⋅=
sT AGu′3 4⁄ sL1 2⁄ λu
1 4⁄– lt1 4⁄
=
A A 0.5=
A 0.6=
G
ε εcr
ε εcr> ε εcr<
ε
G 12--erfc 1
2σ----------- εcr ε⁄( ) σ
2---+ln⎝ ⎠
⎛ ⎞–=
Combustion Theory: Turbulent Burning Velocity
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where denotes the complimentary error function and is the
standard deviation of the distribution of , with being an empirical model coefficient
(default ).
is the thermal conductivity of the unburned mixture. The turbulent flame speed closure
model is completed with the following models for integral velocity fluctuations level:
(Eqn. 60)
integral turbulent length scale:
(Eqn. 61)
and Kolmogorov length scale.
(Eqn. 62)
The critical dissipation rate, , is computed from a specified critical velocity gradient, ,
and the kinematic viscosity of the fluid, , according to:
(Eqn. 63)
For steady laminar flow the critical velocity gradient for quenching, , can be obtained
numerically. However, for turbulent flows, the critical value must be larger than in laminar
cases because the smallest turbulent eddies, which are responsible for the largest strain
rates, do not persist long enough to quench a flame front locally. Furthermore, different
model problems may result in significant variation of the critical values obtained. For these
reasons, the quenching critical velocity gradient has to be tuned for industrial simulations.
In fact, it is the only significant parameter for tuning the model.
Theory or numerical modeling can suggest a range of physically plausible values of . . For
example, the inverse of the chemical time scale of the reaction, , scaled by a factor in the
range 0.1 to 1.0 is a reasonable starting point. For gas turbine combustion chambers
(burning a lean methane/air mixture) values in the range:
to
depending on the configuration, have been used successfully [40] [41] It should be noted
that these recommended values are for atmospheric temperature and pressure.
erfc σ µstr lt η⁄( )ln=
ε µstr
µstr 0.28=
λu
u′ 23--k=
lt k3 2⁄ ε⁄=
η ν3 4⁄ ε1 4⁄⁄=
εcr gcr
ν
εcr 15νgcr2
=
gcr
gcr
τch
gcr 6000 s 1–[ ]= 10000 s 1–[ ]
Combustion Theory: Turbulent Burning Velocity
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Table 4 Default Model Coefficients for the Zimont Turbulent Burning VelocityCorrelation
Peters Correlation
The Peters correlation for turbulent burning velocity was originally developed for the
G-equation combustion model (N. Peters [37]). For ANSYS CFX, it was adopted to be used
with the burning velocity model.
, where , , , and
.
Turbulence effects are modeled as a function of the ratio between the integral turbulence
length scale, , and the laminar flame thickness, .
(Eqn. 64)
(Eqn. 65)
The diffusion coefficient for the flame is computed from molecular properties of the
unburnt mixture indicated by the subscript u. The default values for the model constants are
listed in Table 5.
Table 5 Default Model Coefficients for Peters Turbulent Burning VelocityCorrelation
Parameter Default Value
0.5
0.28
10000
A
µstr
gcr s 1–
Coefficient Default Value
0.37
0.78
2.0
1.0
sT sL 1 σt+( )= σt A llF----– A l
lF----⎝ ⎠
⎛ ⎞ 2 u'lsLlF---------++= u'
23--k= A
a4b32
2b1----------=
B a4b32
=
l lF
l a1u'
3
ε------=
lFDsL----
λ cp⁄( )uρusL
-------------------= =
a1
a4
b1
b3
Combustion Theory: Spark Ignition Model
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Mueller Correlation
The correlation by Mueller et al. follows an approach similar to the Peters correlation. It
accounts for variation of the average laminar burning velocity due to turbulent fluctuations
of mixture fraction and accounts for extinction under high scalar dissipation rates. However,
the Mueller correlation uses a different correlation for turbulence dependency:
, ,
Where is the laminar burning velocity at stiochiometric mixture fraction
(equivalence ratio ). The laminar burning velocity is integrated over a presumed PDF
accounting for fluctuations of mixture fraction.
(Eqn. 66)
The presumed PDF for mixture fraction is described in Coupling of Laminar Flamelet with
the Turbulent Flow Field (p. 237). Table 6 lists the default values for the model coefficients.
Table 6 Default Model Coefficients for Mueller Turbulent Burning VelocityCorrelation
Spark Ignition Model
The purpose of the spark ignition is two-fold: First, it is required in order to provide the
appropriate conditions to start the combustion at time and location of the spark. Second,
the initial size of the spark volume may be too small to be resolved by the mesh. Therefore,
a model is needed in order to describe the initial growth of the spark kernel
The current model assumes that the burnt region around the spark initially grows as a ball.
During this phase the radius of the spark kernel, , is computed solving a zero-dimensional
initial value problem (IVP). The radius at ignition, , is defined by the initial spark
volume, :
(Eqn. 67)
Coefficient Default Value
1.5
0.8
1.0 [1/s]
1.0
sT sL 1 σt+( ) 1 α χχq-----–⎝ ⎠
⎛ ⎞⋅ ⋅= σt b1u'
sL,st-------- b2
u'sL,st--------+= u'
23--k=
sL,st sL Zst( )=
φ 1=
sL P Z( ) sL Z( )⋅( ) Zd0
1∫=
b1
b4
χq
α
rK
t ignite
V initial
tdd rK t( )
ρuρb----- sT,k⋅=
Combustion Theory: Phasic Combustion
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where
The spark kernel radius is mapped onto the three-dimensional flow field by averaging the
reaction progress over the so-called phantom region. The phantom region is a ball of radius
equal to the transition radius, , and center equal to the spark center, . While
solving for the kernel radius, the reaction progress variable is algebraically set:
for
for
The initial value problem is solved until the kernel radius reaches the transition radius,
, specified by the user. At this point, the IVP solver is stopped and transition to the
principal combustion model is made (i.e., switch to the burning velocity model).
The growth rate for the kernel radius is the turbulent burning velocity with a modification
accounting for high curvature while the kernel is small:
(Eqn. 68)
where
The IVP solver uses quantities averaged over the phantom region for laminar and turbulent
burning velocities, and , turbulence quantities, and , and densities of the burnt
and the fresh mixture, and ,.
Phasic Combustion
For multiphase simulations, phasic combustion describes combustion within each
individual phase. The combustion models are solved in exactly the same way as for a single
phase simulation.
• The Eddy Dissipation Model (p. 229)
• The Finite Rate Chemistry Model (p. 230)
• The Combined Eddy Dissipation/Finite Rate Chemistry Model (p. 232)
• The Flamelet Model (p. 233)
• Burning Velocity Model (Premixed or Partially Premixed) (p. 239)
rK t ignite( )3V initial
4π------------------3=
rtrans xspark
crK
rtrans-----------⎝ ⎠
⎛ ⎞ 3= x xspark– rtrans≤
c 0= x xspark– rtrans>
rtrans
sT,k max sL sT2rk----Dt–,⎝ ⎠
⎛ ⎞=
DtCuσt------ k2
ε-----⋅=
sL sT k ε
ρb ρu
Combustion Theory: NO Formation Model
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NO Formation Model
The NO formation model is fully integrated into the ANSYS CFX reaction and combustion
module. This provides complete control to the NO model by means of common interface
(REACTION object) on one hand; on the other hand the generic reaction system profits from
the extensions implemented for the NO model (Arrhenius with Temperature PDF reaction
rates, variable pre-exponential factor for Arrhenius rates).
Formation Mechanisms
The formation of NOx is a complicated process involving several different mechanisms
which are termed:
• Thermal NO
• Prompt NO
• Fuel Nitrogen
• N2O
• Reburn (destruction of NO)
Reactions for the first three formation paths (thermal, prompt and fuel) and for NO reburn
are predefined in the REACTIONS database. It is possible to add reactions for other
mechanisms, or modify the provided mechanisms, using the Reaction details view in
ANSYS CFX-Pre (or manually in the CCL commands file).
Thermal NO The thermal NO mechanism is a predominant source of NOx in gas flames at temperatures
above 1800 K. The NO is formed from the combination of free radical O and N species, which
are in abundance at high temperatures. The two-step mechanism, referred to as the
Zeldovich mechanism, is thought to dominate the process:
(Eqn. 69)
(Eqn. 70)
In sub or near stoichiometric conditions, a third reaction may be important:
(Eqn. 71)
When this step is included with the first two, it is referred to as the extended Zeldovich
mechanism.
O N2 NO N+→+
N O2 NO O+→+
OH N NO H+→+
Combustion Theory: NO Formation Model
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The rates of each of these three reactions (using the subscripts 1,2,3 to describe the three
reactions) are expressed as [43]:
(Eqn. 72)
(Eqn. 73)
(Eqn. 74)
When multiplied by the concentrations of the reactants, they yield rates in terms of
, which can be converted to a volumetric mass source term.
The first step tends to be rate limiting, producing both an NO and N radical species. The N
radical is assumed to be oxidized by reaction (Eqn. 70) in the Zeldovich mechanism and also
by reaction (Eqn. 71) in the extended Zeldovich mechanism. Either way, these second
oxidation reactions are assumed to be fast and if Reaction (Eqn. 69) occurs, then two NO
molecules will be formed. The thermal NO formation in , , is therefore
related to the rate of reaction (Eqn. 69):
(Eqn. 75)
(Eqn. 76)
Here, denotes the molecular mass of NO. Thus, if the molar concentrations [O] and
[N2] of O radicals and N2 are known, the thermal NO mechanism can be calculated.
When using the Laminar Flamelet model, almost always the O radical concentration can be
taken without further assumptions from the solution since the model predicts it directly.
However, when using the Eddy Dissipation model (EDM) and/or the Finite Rate Chemistry
model (FRC), O radical concentrations usually are not known directly but must be derived
from other quantities. Here, the O radical concentration is estimated from the molecular
oxygen dissociation,
(Eqn. 77)
(Westenberg, 1975):
(Eqn. 78)
k1 1.8 1011⋅( ) 38370T
--------------–⎝ ⎠⎛ ⎞exp=
k2 6.4 109⋅( ) 3162T
-----------–⎝ ⎠⎛ ⎞exp=
k3 3.0 1013⋅=
kmol m3⁄ s⁄[ ]
kg m3⁄ s⁄ SNO thermal,
SNO thermal, WNOkthermal O[ ] N2[ ]=
kthermal 2k1=
WNO
12--O2 O⇔
O[ ] 12567 kmol1 2⁄ m 3 2⁄– K 1 2⁄[ ] T 1 2⁄– exp(-31096 K T )⁄ O2[ ]1 2⁄⋅=
Combustion Theory: NO Formation Model
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By substitution, the effective source term for NO then reads:
(Eqn. 79)
(Eqn. 80)
Prompt NO At temperatures lower than 1800 K, hydrocarbon flames tend to have an NO concentration
that is too high to be explained with the Zeldovich mechanisms. Hydrocarbon radicals can
react with molecular nitrogen to form HCN, which may be oxidized to NO under lean flame
conditions.
(Eqn. 81)
(Eqn. 82)
The complete mechanism is very complicated. However, De Soete (see also Peters and
Weber, 1991) proposed a single reaction rate to describe the NO source by the Fenimore
mechanism,
(Eqn. 83)
(Eqn. 84)
and denote molar mass of NO and the mean molar mass of the mixture,
respectively. The Arrhenius coefficients depend on the fuel. (De Soete, 1974) proposed the
following values:
Methane fuel
(Eqn. 85)
Acetylene fuel
(Eqn. 86)
Fuel Nitrogen The fuel nitrogen model assumes that nitrogen is present in the fuel by means of HCN. HCN
is modeled to either form or destroy NO depending on the local conditions in the mixture,
with HCO acting as an intermediate species. The mechanism consists of three reaction steps:
SNO thermal,O2, WNOkthermal O[ ]1 2⁄ N2[ ]=
kthermal,024.52414 1015 m3 2⁄ kmol 1 2⁄– K 1 2⁄ s 1–[ ] T 1 2⁄– exp(69466 K/T)⋅ ⋅⋅=
CH N2 HCN N+→+
HCN O2 NO ...+→+
SNO prompt,
SNO prompt, WNOkprompt O2[ ]1 2⁄ N2[ ] Fuel[ ] Wρ------⎝ ⎠
⎛ ⎞ 3 2⁄=
kprompt Aprompt T A prompt, T⁄–( )exp=
WNO W
Aprompt 6.4 106 1 s⁄[ ]⋅=
T A prompt, 36510 K[ ]=
Aprompt 1.2 106 1 s⁄[ ]⋅=
T A prompt, 30215 K[ ]=
Combustion Theory: NO Formation Model
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1. HCN NO formation:
2. HCN NO destruction:
3. HCO oxidation:
The reaction rates in [mol/s] for the reactions are, respectively:
(Eqn. 87)
where
(Eqn. 88)
(Eqn. 89)
where denotes the molar fraction of and denotes the mean molar mass of the
mixture.
NO Reburn Under fuel rich conditions, when the amount of oxygen available is not sufficient to oxidize
all of the fuel, the excess fuel may lead to reduction of NO. This process can be described by
a global reaction:
(Eqn. 90)
One should keep in mind that this is only a global representation. The physical process
actually occurring is much more complicated. The real process involves many intermediate
components appearing during combustion of the fuel, e.g., CHx radicals, which attack the
NO.
The stoichiometric coefficients for fuel, carbon dioxide and water vapor are fuel-dependent.
For a given fuel they can easily be derived from the element balance. For methane, the
global NO reburn reaction is:
(Eqn. 91)
HCN O2+ HCO NO+→
HCN NO+ HCO N2+→
HCO 34--O2+ CO2
12--H2O+→
R1 3.5 1010 s 1–[ ] HCN[ ] XO2( )α 67kcal
mol----------⎝ ⎠
⎛ ⎞ RT( )⁄–⎝ ⎠⎛ ⎞exp⋅ ⋅ ⋅ ⋅=
α 1
1 XO20.007⁄( )2
+------------------------------------------=
R2 3 1012 s 1–[ ] Wρ------ HCN[ ] NO[ ] 60kcal
mol----------⎝ ⎠
⎛ ⎞ RT( )⁄–⎝ ⎠⎛ ⎞exp⋅ ⋅ ⋅ ⋅=
R3 8.3 106 s 1–[ ] HCO[ ] 0.3– O2[ ]1.3 30kcalmol----------⎝ ⎠
⎛ ⎞ RT( )⁄–⎝ ⎠⎛ ⎞exp⋅ ⋅ ⋅ ⋅=
XO2O2 W
NO νFFuel+12--N2 νCO2
CO2 νH2OH2O+ +→
NO 14--CH4+
12--N2
14--CO2
12--H2O+ +→
Combustion Theory: NO Formation Model
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The reaction rate will be fuel dependent. For coal volatiles, the reaction rate defaults to:
(Eqn. 92)
The same reaction rate is applied for the predefined NO reburn reaction for methane.
TurbulenceEffects
The above reaction rates are applicable to laminar flow, premixed chemistry. In turbulence
systems, fluctuations can have a dominant impact on the NO formation rate. For both the
thermal and prompt NO mechanisms, there is a strong dependence of the rates on the
temperature due to their high activation energy. Thus, temperature fluctuations,
particularly positive fluctuations, can dramatically increase the NO formed in flames. These
temperature fluctuations are included in ANSYS CFX using a statistical approach.
In order to determine the mean rate for NO formation, a presumed probability density
function (presumed PDF) method is used to compute the weighted average of the reaction
rate:
(Eqn. 93)
This integration is carried out separately for each reaction step. For simplicity, the subscripts
(thermal or prompt) have been omitted. The integration range for temperature, , is
the range of temperatures occurring. The default for NO reactions is the range [300 K; 2300
K], but this may be modified by you on a per reaction scope.
The probability density function (PDF) is computed from mean temperature, , and the
variance of temperature, . The shape of is presumed to be that of a beta function
( -function):
(Eqn. 94)
Where:
(Eqn. 95)
and:
(Eqn. 96)
Rreturn 2.72 106 s 1–[ ] Wρ------ NO[ ] Fuel[ ] 9460 K[ ]–( ) T⁄( )exp⋅ ⋅ ⋅ ⋅=
k 1Tu Tl–----------------- k T( ) P T( )• Td
Tl
Tu
∫=
Tl Tu;[ ]
P T
T''2˜
P
β
P T( ) ϕa 1–( ) 1 ϕ–( )b 1–
ψa 1– 1 ψ–( )b 1– ψd01∫
-----------------------------------------------------=
a ϕ ϕ 1 ϕ–( )g
--------------------- 1–⎝ ⎠⎛ ⎞ b, 1 ϕ–( ) ϕ 1 ϕ–( )
g--------------------- 1–⎝ ⎠
⎛ ⎞= =
ϕT Tl–
Tu Tl–----------------- ϕ,
T Tl–
Tu Tl–----------------- g, T''˜
Tu Tl–( )2-------------------------= = =
Combustion Theory: NO Formation Model
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For vanishing temperature variance (vanishing temperature fluctuations), the beta function
is approaching to a single Dirac peak (delta-function). In the limit, the integrated reaction
rate is the same as for that for the standard Arrhenius rate. For very large fluctuation, the
beta function goes towards a double Dirac peak, and for small to medium temperature
variance the shape of the PDF is similar to that of a Gaussian distribution.
Arrhenius reaction rates integrated over a PDF for temperature is not limited to NO
formation but may be used for any reaction.
TemperatureVarianceEquation
For the temperature variance, , that is needed from constructing the probability density
function (PDF) used for the temperature integration, the following transport equation is
solved:
(Eqn. 97)
Default values for the model co-efficients are and . .
The above equation is missing some physical aspects namely the production of
temperature fluctuations due to heat release by chemical reaction. Heat release is
fluctuating, too, because of turbulent fluctuations of the reactants. However, in the current
model temperature variance is only needed as input to another model: The construction of
a probability density function with presumed shape. For this purpose, the above equation
provides sufficient accuracy.
For convenience of setting up a case, the temperature variance equation can be run without
specifying BC or IC data. In the first release, these are not offered by the GUI (ANSYS
CFX-Pre). If absent, zero fluctuations are assumed at inlets, openings and walls with
specified temperature. At walls with specified heat flux or transfer coefficient, the default BC
for temperature variance is zero flux.
Model Control Since the model for NO formation is implemented by means of REACTION objects, you have
full control of all aspects of the model. ANSYS CFX provides the same flexibility for the NO
model as for the generic combustion and reaction system.
Adjusting Model CoefficientsIt is possible for a user to modify any coefficient of the NO model. Since the NO formation
reactions are defined by means of REACTION objects, they may be edited either in ANSYS
CFX-Pre in the CCL Editor or in the CCL text file (commands file).
The model parameter that is most likely to need adjusting is the temperature integration
range for the presumed PDF. It is specified by the Lower Temperature and Upper
Temperature parameters. These appear in the FORWARD REACTION RATE object when
Option is set to Arrhenius with Temperature PDF. The predefined reaction schemes
come with the range set to the interval [300 K, 2300 K]. It is recommended to adjust this to
the maximum temperature range occurring. For many systems, this temperature range is
defined by the minimum inlet temperature and the adiabatic flame temperature.
T''2˜
∂ρT''2˜
∂t---------------
∂ρu j˜ T ''
2˜
∂x j--------------------+
∂∂x j-------- µ
µtPrt-------+
⎝ ⎠⎜ ⎟⎛ ⎞ ∂T ''
2˜
∂x j-----------
⎩ ⎭⎨ ⎬⎧ ⎫
CprodµtPrt------- ∂T˜
∂x j--------
⎝ ⎠⎜ ⎟⎛ ⎞
2
Cdissρεk--T ''
2˜–+=
Cprod 2.0= Cdiss 2.0=
Combustion Theory: Chemistry Post-Processing
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User Defined NO Formation MechanismsThe NO formation model can be extended to a user's specific needs by adding appropriate
reactions for NO, for example, fuel nitrogen or the N2O reaction path. The procedure is the
same as for any other reaction. A user probably may want to chose the Arrhenius with
Temperature PDF option for reactions rate in order to account for turbulent fluctuations of
temperature.
It is also possible to select each NO formation path individually (thermal, prompt, fuel
nitrogen, reburn, or possibly user defined). Simply add only those reactions to the mixture
material that you want to account for.
Chemistry Post-Processing
The components and reactions that are post-processed are one-way coupled to the main
simulation. This means that there is no effect on mixture properties or heat release. The list
of components and reactions for post-processing are specified by the user.
Post-processing components:
• No contribution to mixture properties (density, static enthalpy)
• No contribution to total mass
Post processing reactions:
• Generate sources for components that are also post-processed
• Generate no source for regular components (i.e., not post-processed ones)
• No heat release
The above model assumptions are justified if the component mass fractions and reaction
turnover are small relative to the bulk mixture. A typical application is the simulation of
pollutants, e.g., NO formation. The transport equations for post-processing components are
solved at the end of the simulation after the solving of the main equations. When the
post-processing reactions require a temperature variance equation (reaction rate option is
Arrhenius with Temperature PDF) and the variance equation was not solved during the
main simulation, then temperature variance will be post-processed as well. In this case, the
temperature variance equation will be solved first to convergence or maximum number of
iterations and is then followed by the post-processing components group.
Combustion Theory: Soot Model
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Soot Model
In the Magnussen soot model (Magnussen and Hjertager [46], it is assumed that soot is
formed from a gaseous fuel in two stages, where the first stage represents formation of
radical nuclei, and the second stage represents soot particle formation from these nuclei.
Transport equations are solved for the specific concentration of radical nuclei, [mol/kg],
and for the soot mass fraction, [kg/kg]:
(Eqn. 98)
(Eqn. 99)
The modeling procedure can be grouped into three independent parts:
1. Formation of nuclei and soot particles following the models of Tesner et al., [45]
2. Combustion of nuclei and soot particles
3. Magnussen’s Eddy Dissipation Concept (EDC) for modeling the effect of turbulence on
mean reaction rates.
The soot model can be used in either single phase or multiphase flow (MPF) configurations.
In multiphase calculations, however, the soot variables cannot be a separate phase but must
be assigned to one of the fluids.
Soot Formation
Formation of nuclei and soot particles is computed following the empirical models of Tesner
et al. [45]. The source terms are formulated in terms of particle number concentrations for
nuclei:
(Eqn. 100)
and soot particles:
(Eqn. 101)
Where [part/mol] is Avogadro’s number and:
(Eqn. 102)
XN
Y s˜
∂ ρXN˜( )
∂t-------------------
∂ ρu j˜ XN
˜( )∂x j
-------------------------+ µµtPrt-------+⎝ ⎠
⎛ ⎞ ∂XN˜
∂x j-----------
⎩ ⎭⎨ ⎬⎧ ⎫
Snuclei f, Snuclei c,+ +=
∂ ρY S˜( )
∂t------------------
∂ ρu j˜ Y S
˜( )∂x j
-----------------------+ µµtPrt-------+⎝ ⎠
⎛ ⎞ ∂Y S˜
∂x j---------
⎩ ⎭⎨ ⎬⎧ ⎫
Ssoot f, Ssoot c,+ +=
CN ρ A XN [part/m3]⋅ ⋅=
CS ρY Smp------- [part/m3]⋅=
A 6.02214199 1023⋅=
mp ρsootπd3 6⁄=
Combustion Theory: Soot Model
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is the mass of a soot particle and are the density and the mean diameter of the soot
particles, respectively. With the above definitions, the source terms for nuclei and
soot formation can be modeled as (Tesner et al [45]):
(Eqn. 103)
(Eqn. 104)
In the nuclei equation, the spontaneous formation of radical nuclei from the fuel, , is
modeled using the Arrhenius approach,
(Eqn. 105)
where is the mass fraction of carbon in the fuel material. is a linear branching
coefficient, is a linear termination coefficient, and is a coefficient of linear termination
or radical nuclei on soot particles. In the soot equation, and are constants. The default
values for all of the soot model parameters are summarized in the following table:
Most references list the coefficients using the absolute particle number [part] in the physical
dimensions. However, for numerical, reasons ANSYS CFX is using [mol] instead. To convert
from the [part] system into the [mol] system, coefficients and have to be multiplied by
Avogadro’s number [part/mol], and the coefficient needs to be
divided by . All other coefficients remain unchanged.
Parameter Default Value by [part] Default value by [mol]
Fuel dependent (methane 12/16,acetylene 24/26)
Fuel dependent (methane 12/16,acetylene 24/26)
ρsoot d
Snuclei f,
Ssoot f,
Snuclei f, n0 f g–( )CN g0CN CS–+=
Ssoot f, mp a bCS–( )CN=
n0
n0 a0 f cρY fuel T A 0, T⁄–( )exp=
f c f
g g0
a b
ρsoot 2000[kg/m3] 2000[kg/m3]
d 1.785 10 8– m[ ]⋅ 1.785 10 8– m[ ]⋅
a0 1.35 1037[part/kg/s]⋅ 2.24 1013[mol/kg/s]⋅
f c
T A 0, 90000 K[ ] 90000 K[ ]
f g– 100 1 s⁄[ ] 100 1 s⁄[ ]
g0 1.0 10 15– [m3/s/part]⋅ 6.02 108[m3/s/mol]⋅
a 1.0 105[1/s]⋅ 1.0 105[1/s]⋅
b 8.0 10 14– [m3/s/part]⋅ 4.82 1010[m3/s/mol]⋅
g0 b
A 6.02214199 1023⋅= a0
A
Combustion Theory: Soot Model
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Soot Combustion
The mean rates of combustion of nuclei, , and soot particles, , are calculated
from the fuel consumption reaction rate, in [kg/m3/s], as:
(Eqn. 106)
(Eqn. 107)
This is equivalent to posing the same assumptions to nuclei and soot particles combustion
that were made for fuel consumption. For example, when the Eddy Dissipation model is
applied for the fuel consumption reaction, combustion of nuclei and soot particles is
assumed to operate at the fast chemistry limit. No special model for turbulence effects is
required for the nuclei and soot combustion rates as this is already accounted for in the
computation of the fuel reaction rate.
Turbulence Effects
The above reaction rates are applicable to laminar flow. In turbulent systems, fluctuations
can have a significant impact on nuclei and soot formation because of the non-linearity of
the respective source terms. To account for the effect of turbulence on soot formation, the
Eddy Dissipation Concept (EDC) developed by Magnussen [44] is applied.
The Eddy Dissipation Concept (EDC) is a reactor concept that identifies a reactor, where the
combustion of fuel takes place, related to the fine structures in turbulence. This reactor is
treated as a homogeneous reactor exchanging mass and energy with the surrounding fluid.
The fraction of mass that is contained in the fine structures is determined from the
turbulence quantities.
Local balance equations are solved to compute the temperature and the concentrations of
nuclei and soot particles in the fine structures and in the surrounding fluid, respectively. The
mean source terms are then computed assuming a bimodal distribution of the fluid
between the fine structures and the surrounding. For a detailed explanation of the Eddy
Dissipation Concept, see (Magnussen, 1989).
The EDC procedure need not be applied to the soot combustion terms, because turbulence
is already accounted for in the computation of the fuel reaction rate. For details, see Soot
Combustion (p. 261).
Snuclei c, Ssoot c,
Sfuel
Snuclei c, SfuelXN˜
Y fuel˜
-----------=
Ssoot c, SfuelY S˜
Y fuel˜-----------=
Combustion Theory: Soot Model
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ANSYS CFX-Solver Theory Guide
Radiation Theory
Introduction
The topic(s) in this section include:
• Radiation Transport (p. 263)
• Rosseland Model (p. 268)
• The P1 Model (p. 269)
• Discrete Transfer Model (p. 270)
• Monte Carlo Model (p. 271)
• Spectral Models (p. 272)
This chapter contains a simple summary of the theory of thermal radiation and the
algorithms used in ANSYS CFX. Details on modeling radiation in ANSYS CFX are available.
For details, see Radiation Modeling (p. 289 in "ANSYS CFX-Solver Modeling Guide").
Radiation Transport
The goal of radiation modeling is to solve the radiation transport equation, obtain the
source term, S, for the energy equation, and the radiative heat flux at walls, among others
quantities of interest. You should restrict yourself to coherent time-independent radiation
processes. This is normally a very good approximation of situations likely to be met in
industrial applications because the time scale for radiation to come into local equilibrium is
very short and the temperatures are relatively low.
The spectral radiative transfer equation (RTE) can be written as:
(Eqn. 1)
dIν r s,( )ds
---------------------
Kaν Ksν+( )Iν r s,( )– KaνIb ν T,( )Ksν4π-------- dIν r s',( )Φ s s'•( ) Ω' S+d
4π∫+ +
⎝ ⎠⎜ ⎟⎛ ⎞
=
Radiation Theory: Radiation Transport
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where:
• v = frequency
• r = position vector
• s = direction vector
• s = path length
• Ka = absorption coefficient
• Ks = scattering coefficient
• = Blackbody emission intensity
• = Spectral radiation intensity which depends on position (r) and direction (s)
• T = local absolute temperature
• = solid angle
• = in-scattering phase function
• S = radiation intensity source term
The RTE is a first order integro-differential equation for Iv in a fixed direction, s. To solve this
equation within a domain, a boundary condition for Iv is required. The following are the
boundary conditions currently supported in ANSYS CFX:
• Diffusely emitting and reflecting opaque boundaries
(Eqn. 2)
where =spectral emissivity.
• Diffusely emitting and specularly reflecting boundaries
(Eqn. 3)
where:
• =diffuse reflectivity= *diffuse fraction
• =specular reflectivity= *(1-diffuse fraction)
• =spectral reflectivity= =
• =specular direction
• Semi-transparent walls (Monte Carlo only)
Due to the dependence on 3 spatial coordinates, 2 local direction coordinates, s, and
frequency, the formal solution of the radiative transfer equation is very time consuming and
usually accomplished by the use of approximate models for the directional and spectral
Ib
Iν
Ω
Φ
Iν rw s,( ) εν rw( )Ib ν T,( )ρw rw( )
π----------------- Iν rw s',( ) n s'• Ω'd
n s' 0<•∫+=
εν
Iν rw s,( )
εν rw( )Ib ν T,( )ρν
d rw( )π
---------------- Iν rw s',( ) n s'• Ω'dn s' 0<•
∫ ρνs rw( )Iν rw ss,( )++
⎝ ⎠⎜ ⎟⎛ ⎞
=
ρνd 1 εν–( )
ρνs 1 εν–( )
ρν ρνd ρ+ ν
s1 εν–( )
ss
Radiation Theory: Radiation Transport
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dependencies. For directional approximations, ANSYS CFX includes Rosseland, P-1,
Discrete Transfer and Monte Carlo. For spectral approximations, ANSYS CFX includes:
Gray, Multiband and Weighted Sum of Gray Gases.
Blackbody Emission
The energy spectrum for radiation emitted by a blackbody is represented by:
(Eqn. 4)
where:
• = refractive index
• = Planck’s constant
• = speed of light in vacuum
• = Boltzmann’s constant
Writing , you have:
(Eqn. 5)
where = Stefan-Boltzmann constant:
(Eqn. 6)
The total blackbody emission is simply the integral of over all frequencies:
(Eqn. 7)
Note that the blackbody emission is proportional to the fourth power of the temperature
and because of this strong dependence, radiation is usually unimportant or totally
dominant for heat transfer.
The sun, for example, is approximately a blackbody at a temperature of 5700 K. The
spectrum peaks in the yellow part of the visible spectrum.
Eb ν T,( ) 2πv2
c2----------- n2hv
ehv kBT⁄( )
1–------------------------------ W m 2– Hz 1–[ ] πIb ν T,( )= =
n
h
c
kB
x hv kBT⁄=
Eb x T,( ) n2σT4 hkT-------⎝ ⎠
⎛ ⎞ 15
π4----- x3
ex 1–-------------
⎝ ⎠⎜ ⎟⎛ ⎞
=
σ
σ 2π5k4
15h3c2----------------≡
Eb
Eb T( ) Eb x T,( ) xd0
∞
∫ n2σT 4= =
Radiation Theory: Radiation Transport
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Combustion temperatures are typically 1000 - 2000 K, with spectrum peaks in the near
infra-red range. Note that the peak of the spectrum as a function of wavelength is at:
(Eqn. 8)
Quantities of Interest
The spectral radiative heat flux, , passing through a surface at some location r with a unit
vector normal n is:
(Eqn. 9)
Integrating the equation of transfer over solid angles, the divergence of the spectral
radiative heat flux is given by:
(Eqn. 10)
where Gv is the spectral incident radiation, given by:
(Eqn. 11)
The total radiative flux is obtained by integrating (Eqn. 10) over the spectrum:
(Eqn. 12)
In the case of pure scattering, . Therefore , as it should since in this case
no energy is lost from the radiation field; clearly this is also true in thermodynamic
equilibrium.
OpticalThickness
Optical thickness is a measure of the ability of a given path length of gas to attenuate
radiation of a given wavelength. Optical thickness is given by:
(Eqn. 13)
where is the optical thickness (or opacity) of the layer of thickness and is a function
of all the values of between 0 and . A large optical thickness means large attenuation.
hv 4kBT∼
qv
qνR r n,( ) s n•( )Iν r s,( ) Ωsd∫=
qνR∇•–( ) Ka Gν 4Ebν–( )=
Gν Iν Ωd s∫≡
qR∇• qνR νd
0
∞∫ K avGv vd
0
∞∫ 4 K avEbν vd
0
∞∫–= =
K a 0= qR∇• 0=
κλ S( ) Kλ S*( ) S*d0
S∫=
κλ S( ) S
Kλ S
Radiation Theory: Radiation Transport
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Radiation Through Domain Interfaces
If radiation is included through conducting solids, then usually the difference in refractive
indices between the fluid and solid determines the amount of reflection and refraction that
occurs. The probability of being reflected is given by Fresnels’ equation
(Eqn. 14)
The fraction of the electromagnetic wave that is reflected normally depends on the
polarization of the photon. ANSYS CFX does not keep track of photon polarizations.
Assuming that the photons are randomly polarized, then it is sufficient to take the average
result. The two extreme polarizations are termed transverse electric (TE) and transverse
magnetic (TM), and describe the orientations of the electric and magnetic vectors relative to
the interface.
For the TE mode, the ratio of reflected to incident wave amplitude (EE) is given by:
(Eqn. 15)
and for the TM mode the ratio of reflected to incident wave amplitude ( ) is given by:
(Eqn. 16)
where and are the incident and refracted angles, and and are the refractive
indices of the two media.
The probability of being reflected is determined by the energy flow at the interface which is
proportional to the square of the wave amplitude. Thus, the average reflection coefficient is
given as:
(Eqn. 17)
and the probability of being transmitted is:
(Eqn. 18)
ρ 12--
tan2 θ1 θ2–( )
tan2 θ1 θ2+( )--------------------------------
sin2 θ1 θ2–( )
sin2 θ1 θ2+( )--------------------------------+=
EE
n1n2----- θcos φcos–
n1n2----- θcos φcos+
------------------------------------=
EM
EM
n1n2----- φ θcos–cos
n1n2----- φ θcos+cos------------------------------------=
θ φ n1 n2
0.5 EE2 EM
2+( )
1 0.5 EE2 EM
2+( )–
Radiation Theory: Rosseland Model
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No absorption takes place at the interface, so the probability of transmission plus reflection
is always one. If the photon is transmitted, then the angle of refraction is determined by
Snells’ law:
(Eqn. 19)
ANSYS CFX performs these calculations at every radiation element boundary, although, in
most cases, there is no change of refractive index.
Rosseland Model
The Rosseland approximation is a simplification of the Radiative Transport Equation (RTE)
for the case of optically thick media. It introduces a new diffusion term into the original
energy transport equation with a strongly temperature-dependent diffusion coefficient.
A good source for the simplification of the Radiation Transport Equation for the optically
thick limit can be seen in Siegel and Howe [23]. The total radiative heat flux in an optically
thick, and linearly anisotropic scattering medium can be written as:
(Eqn. 20)
where is the extinction coefficient (i.e., absorption plus scattering).
When the Rosseland Approximation is introduced into the energy transport equation, the
conduction and radiative heat flux can be combined as:
(Eqn. 21)
(Eqn. 22)
(Eqn. 23)
where is the thermal conductivity and is the “total radiative conductivity.” (Eqn. 21) is
called upon to calculate the temperature field in the energy equation.
φsinθsin
-----------n2n1-----=
qr4
3β CKs–----------------------∇Ebv vd
0
∞∫–=
β
q qc qr+=
k kr+( )∇T–=
where kr16σn2T3
3β----------------------–=
k kr
Radiation Theory: The P1 Model
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Wall Treatment
The Rosseland approximation is not valid near walls. Therefore, a special boundary
condition must be specified when heat conduction is comparable to radiation heat transfer.
It has been proposed [24] that a temperature slip boundary condition should be in this
region. From [24] the heat flux at the wall, is given by:
(Eqn. 24)
where is the slip co-efficient, is the wall temperature and is the gas temperature
at the wall. is computed as the solution of:
(Eqn. 25)
where is given by:
(Eqn. 26)
The P1 Model
The Differential Approximation or P1 is also a simplification of the Radiation Transport
Equation, which assumes that the radiation intensity is isotropic or direction independent
at a given location in space. The full form of the radiant energy equation and the derivation
of the P1 model for radiation are given in Raithby [8]. Only a brief summary will be given
here.
The spectral radiative heat flux in the diffusion limit for an emitting, absorbing, and linearly
scattering medium, can be computed as:
(Eqn. 27)
qr w,
qr w,σ Tw
4 T g4
–( )–
ψ-------------------------------=
ψ Tw T g
ψ
ψ 34π------ 1
φv-----atan φd
0
1
∫=
φv
φv1π--- βk
8φ3σT3w
--------------------- 2φ---– 1 φ–
1 φ+------------ln–=
qrv1
3 K av Ksv–( ) AK sv–-------------------------------------------------- Gν∇–=
Radiation Theory: Discrete Transfer Model
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The equation for the spectral incident radiation that results from substituting the above
terms into the radiation transport equation:
(Eqn. 28)
where is the linear anisotropy coefficient.
Wall Treatment
Assuming that the radiation intensity arriving at and leaving a wall are directionally
independent, the boundary condition for (Eqn. 28) at walls is:
(Eqn. 29)
where is the unit vector outward normal to the wall, is a distance coordinate in the
same direction, and represents the value at the wall.
Discrete Transfer Model
The implementation of the Discrete Transfer model in ANSYS CFX assumes that the
scattering is isotropic; therefore, (Eqn. 1) can be simplified as:
(Eqn. 30)
Assuming that the system is reasonably homogeneous, so that:
(Eqn. 31)
the approach is then to solve for the intensity, , along rays leaving from the boundaries
using the equation of transfer:
(Eqn. 32)
where:
= Radiation Intensity leaving the boundary
= Mean Radiation Intensity
13 K av K sv–( ) AKsv–-------------------------------------------------- Gv∇⎝ ⎠
⎛ ⎞∇•– Kav Ebv Gv–( )=
A
n q⋅ rv1
3 K av K sv–( ) AK sv–--------------------------------------------------
∂Gv
∂n+---------–
εv2 2 εv–( )--------------------- Ebv Gv–[ ]
w= =
n n+
w
dIν r s,( )ds
--------------------- K aν K sν+( )Iν r s,( )– K aIb ν T,( )K sν4π-------- Iν r s',( ) Ω' S+d
4π∫+ +=
Iν r( ) Iν r dr+( )∼ qνR r( ) qν
R r dr+( )∼
Iν
Iν r s,( ) Iνo K aν K sν+( )s–( )exp Ibv 1 Kas–( )exp–( ) K sνIv+ +=
Iνo
Iν
Radiation Theory: Monte Carlo Model
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Then, integrate over solid angle at discrete points to get the spectral incident radiation,
and the radiative heat flux, and use the homogeneity assumption to extend the solution
to the entire domain. Non-linearities in the system due to scattering, diffuse reflection, or
temperature dependency of radiation quantities is overcome by iteration.
Since the objective of thermal radiation modeling is to obtain the total volumetric
absorption and emission, additional calculations are still needed. For the Gray spectral
model, the calculation is done once for a unique radiation intensity field. For the Multiband
and Multigray/Weighted Sum of Gray Gases, the solution must be computed for each
spectral band/ gray gas and a final integration to obtain the total radiation quantities is
required. Under the assumption of coherent radiation field, ie., the solution at a given
frequency is independent of that at all other frequencies.
Monte Carlo Model
The Monte Carlo model assumes that the intensity is proportional to the differential angular
flux of photons and you can think of the radiation field as a photon gas. For this gas, is
the probability per unit length that a photon is absorbed at a given frequency. Therefore,
the mean radiation intensity, is proportional to the distance traveled by a photon in unit
volume at , in unit time.
Similarly is proportional to the rate of incidence of photons on the surface at , since
volumetric absorption is proportional to the rate of absorption of photons.
By following a typical selection of photons and tallying, in each volume element, the
distance traveled, you can obtain the mean total intensity.
By following a typical selection of photons and tallying, in each volume element, the
distance times the absorption coefficient, you can obtain the mean total absorbed intensity.
By following a typical selection of photons and tallying, in each volume element, the
distance times the scattering coefficient, you can obtain the mean total scattered intensity.
By also tallying the number of photons incident on a surface and this number times the
emissivity, you obtain the mean total radiative flux and the mean absorbed flux.
Note that no discretization of the spectrum is required since differential quantities are not
usually important for heat transfer calculations. Providing that the spectral (Multiband or
Multigray) selection is done properly, the Monte Carlo tallying automatically integrates over
the spectrum.
I G
qr
Ka
I
r
qvR r
Radiation Theory: Spectral Models
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Spectral Models
The radiation intensity field is also a function of the spectrum as shown in (Eqn. 1). In order
to make the spectral dependence tractable in conjunction with a flow calculation, ANSYS
CFX supports three different models for the spectral dependency of the radiative transfer
equation: Gray, Multiband and Multigray/Weighted Sum of Gray Gases.
Gray
The Gray model assumes that all radiation quantities are nearly uniform throughout the
spectrum, consequently the radiation intensity is the same for all frequencies. Then, the
dependency of (Eqn. 1) on frequency can be dropped.
This implies that only one radiative transfer equation must be solved and that all total
radiation quantities and their spectral counterpart are the same.
Multiband Model
For this model, the spectrum is sub-divided into spectral bands of finite width where
radiative quantities are nearly uniform or can be averaged without losing accuracy. These
bands should span the thermal radiation section of the spectrum. It is assumed that the
value at a given spectral band is represented by the spectral band midpoint value in
frequency domain.
ANSYS CFX assumes that the main spectral variable is frequency, since it is independent of
the material refractive index and it will facilitate the setup of multidomain problems. Other
spectral variables, such wavelength and wavenumber would be available for vacuum only.
Then, the radiative transfer equation is integrated within is spectral band and a modified
RTE is obtained:
(Eqn. 33)
for , where the emission within the spectral band is weighted by:
(Eqn. 34)
After solving one RTE per spectral band, total radiation intensity can be computed as:
(Eqn. 35)
N
d∆νI∆ν r s,( )ds
------------------------------- Kaν Ksν+( )∆νI∆ν r s,( )– KaF∆νIb T( )
Ksν4π-------- ∆νI∆ν r s',( )Φ s s'•( ) Ω' S+d
4π∫
+ +=
∆νI∆ν
F∆ν Eb ν T,( ) νdν1
ν2
∫ Eb ν T,( ) νd0
ν2
∫ Eb ν T,( ) νd0
ν1
∫–= =
I r s,( ) ∆νI∆ν r s,( )1
N
∑=
Radiation Theory: Spectral Models
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This immediately suggests that for an -band model, times as much work is required as
for a grey, -band model. In the case of the Discrete Transfer model, for small this turns
out not to be true because the tracking of the rays through the geometry is a major one-off
overhead.
This model can be used in conjunction with all available radiation models.
Multigray Model
The radiative absorption and emission from a gas can be characterized by the emissivity as
a function of temperature and pL, that is the product of the partial pressure and the path
length. In the context of typical combustion systems, the dominant emitters of radiation are
carbon dioxide and water vapor (although hydrocarbons, CO and SO2 also make a minor
contribution). Hottel and Sarofim [48] have published emissivity charts for CO2 and H2O that
have been obtained by a combination of measurement and extrapolation. These plots show
that emissivity is strongly dependent on and also has a weaker dependence on the gas
temperature. This functional dependence can be accurately correlated by assuming that the
emissivity arises as the result of independent emission from a sufficient number of gray
gases:
(Eqn. 36)
Since emissivity must be proportional to absorptivity by Kirchoffs’ law, it follows that
must approach unity as . This imposes a constraint on the gray gas weights or
amplitudes:
(Eqn. 37)
Also the requirement that is a monotonically increasing function of is satisfied if all
the are positive.
If the number of grey gases, , is large, then may be thought of as the fraction of the
energy spectrum, relative to the blackbody energy, for which the absorption coefficient is
approximately . Then, the methodology described for the Multiband model can be used
directly.
Multi-Grey GasModelParameters
Hadvig [49] has published charts of emissivity of combined CO2-H2O mixtures, for mixtures
with different relative proportions of CO2 and H2O. For the case of natural gas combustion,
it can be shown that the proportions of water vapor and carbon dioxide in the products of
N N
I N
pL
εg agi 1 ekiLp–
–( )i 1=
N g
∑=
εg
pL ∞→
agii 1=
N g
∑ 1=
εg pL
agi
N g agi
ki
Radiation Theory: Spectral Models
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combustion is such that partial pressure ratio, / is approximately equal to 2.
Similarly, this ratio is 1 for oils and other fuels with the empirical formula, (CH2)x. Most other
hydrocarbon fuels have combustion products with a / ratio lying between 1
and 2. Starting from the charts of Hottel and Sarofim (1967) [48] for CO2 and H2O and
applying their correction factor for mixtures, Hadvig has evaluated the emissivity of a gas
mixture with / = 1 and 2 and presented the results as a function of and .
Leckner [50] has also published emissivity data, based on integrating the measured spectral
data for CO2 and H2O, which is in reasonable agreement with the Hottel charts where the
charts are based on measured data.
Taylor and Foster (1974) [51] have integrated the spectral data and constructed a multigrey
gas representation:
(Eqn. 38)
where the are represented as linear functions of :
(Eqn. 39)
As well as CO2 and H2O, the model developed by Beer, Foster and Siddall [52] takes into
account the contribution of CO and unburnt hydrocarbons, e.g., methane (CH4) which are
also significant emitters of radiation. These authors generalize the parameterization of the
absorption coefficients as follows:
(Eqn. 40)
where is the partial pressure of CO and is the total partial pressure of all
hydrocarbon species.
The values of , [K-1], [m-1 atm-1] and [m-1 atm-1] are given in Table 1,
together with a similar correlation for = 3, derived by Beer, Foster and Siddall [52], and
suitable defaults for = 2 or 1 (single gray gas) representations.
pH2O pCO2
pH2O pCO2
pH2O pCO2T g pL
εg agi T g( ) 1 eK i pH2O pCO2
+( )L––
i 1=
4
∑=
agi T g
agi b1i 10 5– b2iT g+=
Ki pH2O pCO2+( ) K i pH2O pCO2
pCO+ +( ) K HCi pHC+→
pCO pHC
b1i b2i K i K HCi
N g
N g
Radiation Theory: Spectral Models
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Table 1 Grey gas emissivity parameters for a carbon dioxide / water vapor /hydrocarbon mixture.
Note: To satisfy the requirement that the factors sum to unity, the factors must sum
to 1.0 and the factors must sum to 0.
Ng i Gaseous Fuels pH2O/pCO2 = 2 Oils pH2O/pCO2 = 1
b1i b2i ki kHCi b1i b2i ki kHCi
1 1 1 0 1 0 1 0 1 0
2 1 0.437 7.13 0 3.85 0.486 8.97 0 3.41
2 0.563 -7.13 1.88 0 0.514 -8.97 2.5 0
3 1 0.437 7.13 0 3.85 0.486 8.97 0 3.41
2 0.390 -0.52 1.88 0 0.381 -3.96 2.5 0
3 1.173 -6.61 68.83 0 0.133 -5.01 109 0
4 1 0.364 4.74 0 3.85 0.4092 7.53 0 3.41
2 0.266 7.19 0.69 0 0.284 2.58 0.91 0
3 0.252 -7.41 7.4 0 0.211 -6.54 9.4 0
4 0.118 -4.52 80 0 0.0958 -3.57 130 0
ai b1
b2
Radiation Theory: Spectral Models
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ANSYS CFX-Solver Theory Guide
Discretization and SolutionTheory
Introduction
This chapter describes:
• Numerical Discretization (p. 277)
• Solution Strategy - The Coupled Solver (p. 292)
• Discretization Errors (p. 296)
Numerical Discretization
Analytical solutions to the Navier-Stokes equations exist for only the simplest of flows under
ideal conditions. To obtain solutions for real flows, a numerical approach must be adopted
whereby the equations are replaced by algebraic approximations which may be solved
using a numerical method.
Discretization of the Governing Equations
This approach involves discretizing the spatial domain into finite control volumes using a
mesh. The governing equations are integrated over each control volume, such that the
relevant quantity (mass, momentum, energy, etc.) is conserved in a discrete sense for each
control volume.
Discretization and Solution Theory: Numerical Discretization
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The figure below shows a typical mesh with unit depth (so that it is two-dimensional), on
which one surface of the control volume is represented by the shaded area.
Figure 1 Control Volume Surface
It is clear that each node is surrounded by a set of surfaces that define the control volume.
All the solution variables and fluid properties are stored at the element nodes.
Consider the mean form of the conservation equations for mass, momentum and a passive
scalar, expressed in Cartesian coordinates:
(Eqn. 1)
(Eqn. 2)
(Eqn. 3)
Node
Element Face Center
Finite Volume surface
Element
∂ρ∂t------
x j∂∂ ρU j( )+ 0=
∂∂t----- ρU i( )
x j∂∂ ρU jU i( )+ ∂P
∂xi--------–
x j∂∂ µeff
U i∂x j∂
---------U j∂xi∂
----------+⎝ ⎠⎜ ⎟⎛ ⎞
⎝ ⎠⎜ ⎟⎛ ⎞
+=
∂∂t----- ρφ( )
x j∂∂ ρU jφ( )+
x j∂∂ Γeff
φ∂x j∂
--------⎝ ⎠⎛ ⎞
⎝ ⎠⎛ ⎞ Sφ+=
Discretization and Solution Theory: Numerical Discretization
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These equations are integrated over a control volume, and Gauss’ Divergence Theorem is
applied to convert some volume integrals to surface integrals. If control volumes do not
deform in time, then the time derivatives can be moved outside of the volume integrals and
the equations become:
(Eqn. 4)
(Eqn. 5)
(Eqn. 6)
where V and s respectively denote volume and surface regions of integration, and dnj are
the differential Cartesian components of the outward normal surface vector. The volume
integrals represent source or accumulation terms, and the surface integrals represent the
summation of the fluxes. Note that changes to these equations due to control volume
deformation are presented below. For details, see Mesh Deformation (p. 290).
The first step in the numerical solution of these exact differential equations is to create a
coupled system of linearized algebraic equations. This is done by converting each term into
a discrete form. Consider, for example, an isolated mesh element like the one shown below.
Figure 2 Mesh Element
Volumetric (i.e., source or accumulation) terms are converted into their discrete form by
approximating specific values in each sector and then integrating those values over all
sectors that contribute to a control volume. Surface flow terms are converted into their
discrete form by first approximating fluxes at integration points, ipn, which are located at
ddt----- ρ Vd
V∫ ρU j n jd
s∫+ 0=
ddt----- ρU i Vd
V∫ ρU jU i n jd
s∫+ P n jd
s∫– µeff
U i∂x j∂
---------U j∂xi∂
----------+⎝ ⎠⎜ ⎟⎛ ⎞
n jds∫ SU i
VdV∫+ +=
ddt----- ρφ Vd
V∫ ρU jφ n jd
s∫+ Γeff
φ∂x j∂
--------⎝ ⎠⎛ ⎞ n jd
s∫ Sφ Vd
V∫+=
n3
Integration Point
Element Face Center
Sectors
ip1
ip2ip3
n1 n2
Discretization and Solution Theory: Numerical Discretization
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the center of each surface segment in a 3D element surrounding the control volume. Flows
are then evaluated by integrating the fluxes over the surface segments that contribute to a
control volume.
Many discrete approximations developed for CFD are based on series expansion
approximations of continuous functions (such as the Taylor series). The order-accuracy of
the approximation is determined by the exponent on the mesh spacing or timestep factor
of the largest term in the truncated part of the series expansion. This is often the first term
excluded from the approximation. Increasing the order-accuracy of an approximation
generally implies that errors are reduced more quickly with mesh or timestep size
refinement. Unfortunately, in addition to increasing the computational load, high-order
approximations are also generally less robust (i.e., less numerically stable) than their
low-order counterparts.
The discrete form of the integral equations becomes:
(Eqn. 7)
(Eqn. 8)
(Eqn. 9)
where V is the control volume, is the timestep, is the discrete outward surface
vector, the subscript ip denotes evaluation at an integration point, and summations are over
all the integration points of the control volume. Note that the First Order Backward Euler
scheme has been assumed in this equation, although a second order scheme is also
available as discussed below. The superscript o refers to the old time level. The discrete mass
flow through a surface of the control volume, denoted by , is given by:
(Eqn. 10)
Pressure-VelocityCoupling
ANSYS CFX uses a co-located (non-staggered) grid layout such that the control volumes are
identical for all transport equations. As discussed by Patankar [118], however, naïve
co-located methods lead to a decoupled (checkerboard) pressure field. Rhie and Chow [2]
V ρ ρo–∆t
--------------⎝ ⎠⎛ ⎞ ρU j∆n j( )ip
ip∑+ 0=
VρU i ρoU i
o–
∆t------------------------------
⎝ ⎠⎜ ⎟⎛ ⎞
mip U i( )ipip∑+ P∆ni( )ip +
ip∑=
µeffU i∂x j∂
---------U j∂xi∂
----------+⎝ ⎠⎜ ⎟⎛ ⎞
∆n j⎝ ⎠⎜ ⎟⎛ ⎞
ipip∑ SU i
V+
V ρφ ρoφo
–∆t
------------------------⎝ ⎠⎛ ⎞ mipφip
ip∑+ Γeff
φ∂x j∂
--------∆n j⎝ ⎠⎛ ⎞
ipip∑ SφV+=
∆t ∇n j
mip
mip ρU j∆n j( )ip=
Discretization and Solution Theory: Numerical Discretization
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proposed an alternative discretization for the mass flows to avoid the decoupling, and this
discretization was modified by Majumdar [119] to remove the dependence of the
steady-state solution on the timestep.
A similar strategy is adopted in ANSYS CFX. By applying a momentum-like equation to each
integration point, the following expression for the advecting (mass-carrying) velocity at
each integration point is obtained:
(Eqn. 11)
(Eqn. 12)
(Eqn. 13)
= approximation to central coefficient of momentum equation, excluding the transient
term
(Eqn. 14)
The overbars indicate averaging of adjacent vertex values to the integration point, while theo superscript denotes values at the previous timestep.
The naïve discretization, given simply by averaging the adjacent vertex velocities to the
integration point, is augmented by a high order pressure variation that scales with the mesh
spacing. In particular, when substituted into the continuity equation, the expression
becomes a fourth derivative of pressure which scales with . This expression
represents term that is spatially third-order accurate, and is sometimes also called the
pressure redistribution term. This term is usually significantly smaller than the average of
vertex velocities, especially as the mesh is refined to reasonable levels.
In some cases, the pressure redistribution term can produce apparently significant spurious
velocity fields. This may occur when a strong pressure gradient is required to balance a body
force, , such as buoyancy or porous drag. In these cases, the Rhie Chow discretization may
U i ip, U i ip, f ipp∂xi∂
-------ip
p∂xi∂
-------ip
–⎝ ⎠⎜ ⎟⎛ ⎞
cip f ip U i ip,o U i ip,
o–( )–+=
f ipdip
1 cipdip–----------------------=
dipVA----–=
A
cipρ∆t------=
f ipp∂xi∂
-------ip
p∂xi∂
-------ip
–⎝ ⎠⎜ ⎟⎛ ⎞
∆x( )3
Si
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lead to velocity wiggles at locations where the body force is discontinuous (e.g., at a free
surface interface, or at the boundary of a porous region). The wiggles are greatly reduced or
eliminated by redistributing the body force as follows:
(Eqn. 15)
Transient Term For control volumes that do not deform in time, the general discrete approximation of the
transient term for the nth timestep is:
(Eqn. 16)
where values at the start and end of the timestep are assigned the superscripts n+ and n-,
respectively.
With the First Order Backward Euler scheme, the start and end of timestep values are
respectively approximated using the old and current time level solution values. The
resulting discretization is:
(Eqn. 17)
It is robust, fully implicit, bounded, conservative in time, and does not yield a timestep size
limitation. This discretization is, however, only first-order accurate in time and will introduce
discretization errors that tend to diffuse steep temporal gradients. This behavior is similar
to the numerical diffusion experienced with the Upwind Difference Scheme for discretizing
the advection term.
With the Second Order Backward Euler scheme, the start and end of timestep values are
respectively approximated as:
(Eqn. 18)
(Eqn. 19)
f ipp∂xi∂
------- Si–⎝ ⎠⎛ ⎞
ip
p∂xi∂
------- Si–⎝ ⎠⎛ ⎞
ip–⎝ ⎠
⎛ ⎞
t∂∂ ρφ Vd
V∫ V ρφ( )
n 12--+
ρφ( )n 1
2--–
–∆t
------------------------------------------------≈
t∂∂ ρφ Vd
V∫ V ρφ ρoφo
–∆t
------------------------⎝ ⎠⎛ ⎞=
ρφ( )n 1
2--–
ρφ( )o 12-- ρφ( )o ρφ( )oo
–( )+=
ρφ( )n 1
2--+
ρφ( ) 12-- ρφ( ) ρφ( )o
–( )+=
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When these values are substituted into the general discrete approximation, (Eqn. 16), the
resulting discretization is:
(Eqn. 20)
This scheme is also robust, implicit, conservative in time, and does not create a timestep
limitation. It is second-order accurate in time, but is not bounded and may create some
nonphysical solution oscillations. For quantities such as volume fractions, where
boundedness is important, a modified Second Order Backward Euler scheme is used
instead.
Shape Functions The solution fields are stored at the mesh nodes. However, various terms in the equations
require solutions or solution gradients to be approximated at integration points. Finite
element shape functions are consequently used to evaluate the solution and its variation
within mesh elements.
A variable varies within an element as follows:
(Eqn. 21)
where Ni is the shape function for node i and is the value of at node i. The summation
is over all nodes of an element. Key properties of shape functions include:
(Eqn. 22)
(Eqn. 23)
The shape functions used in ANSYS CFX are linear in terms of parametric coordinates. They
are used to calculate various geometric quantities as well, including ip coordinates and
surface area vectors. This is possible because (Eqn. 21) also holds for the coordinates:
(Eqn. 24)
The trilinear shape functions for each supported mesh element are given below:
t∂∂ ρφ Vd
V∫ V 1
∆t------ 3
2-- ρφ( ) 2 ρφ( )o
–12-- ρφ( )oo
+⎝ ⎠⎛ ⎞≈
φ
φ Niφii 1=
N node
∑=
φi φ
Nii 1=
N node
∑ 1=
At node j, Ni1 i j=
0 i j≠⎩⎨⎧
=
y Ni yii 1=
N node
∑=
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Hexahedral ElementFigure 3
The tri-linear shape functions for the nodes are:
(Eqn. 25)
s
t
u
3
2
6
5
1
8
4
7
N1 s t u, ,( ) 1 s–( ) 1 t–( ) 1 u–( )=
N2 s t u, ,( ) s 1 t–( ) 1 u–( )=
N3 s t u, ,( ) st 1 u–( )=
N4 s t u, ,( ) 1 s–( )t 1 u–( )=
N5 s t u, ,( ) 1 s–( ) 1 t–( )u=
N6 s t u, ,( ) s 1 t–( )u=
N7 s t u, ,( ) stu=
N8 s t u, ,( ) 1 s–( )tu=
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Tetrahedral ElementFigure 4
The tri-linear shape functions for the nodes are:
(Eqn. 26)
Wedge ElementFigure 5
s
t
u
3
21
4
N1 s t u, ,( ) 1 s– t– u–=
N2 s t u, ,( ) s=
N3 s t u, ,( ) t=
N4 s t u, ,( ) u=
s
t
u
3
21
4
5
6
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The tri-linear shape functions for the nodes are:
(Eqn. 27)
Pyramid ElementFigure 6
The tri-linear shape functions for the nodes are:
(Eqn. 28)
At this point, it is worth highlighting that these shape functions yield linear combinations of
positively and negatively weighted nodal values. The significance of these positive and
negative influences is identified in Discretization Errors (p. 296).
N1 s t u, ,( ) 1 s– t–( ) 1 u–( )=
N2 s t u, ,( ) s 1 u–( )=
N3 s t u, ,( ) t 1 u–( )=
N4 s t u, ,( ) 1 s– t–( )u=
N5 s t u, ,( ) su=
N6 s t u, ,( ) tu=
s
t
u
4
21
5
3
N1 s t u, ,( ) 1 s–( ) 1 t–( ) 1 u–( )=
N2 s t u, ,( ) s 1 t–( ) 1 u–( )=
N3 s t u, ,( ) st 1 u–( )=
N4 s t u, ,( ) 1 s–( )t 1 u–( )=
N5 s t u, ,( ) u=
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Diffusion Terms Following the standard finite element approach, shape functions are used to evaluate
spatial derivatives for all the diffusion terms. For example, for a derivative in the x direction
at integration point ip:
(Eqn. 29)
The summation is over all the shape functions for the element. The Cartesian derivatives of
the shape functions can be expressed in terms of their local derivatives via the Jacobian
transformation matrix:
(Eqn. 30)
The shape function gradients can be evaluated at the actual location of each integration
point (i.e., true tri-linear interpolation), or at the location where each ip surface intersects the
element edge (i.e., linear-linear interpolation). The latter formulation improves solution
robustness at the expense of locally reducing the spatial order-accuracy of the discrete
approximation.
PressureGradient Term
The surface integration of the pressure gradient in the momentum equations involves
evaluating of the expression:
(Eqn. 31)
The value of Pip is evaluated using the shape functions:
(Eqn. 32)
As with the diffusion terms, the shape function used to interpolate P can be evaluated at the
actual location of each integration point (i.e., true tri-linear interpolation), or at the location
where each ip surface intersects the element edge (i.e., linear-linear interpolation). As in the
discretization of the diffusion terms, the formulation improves solution robustness but
reduces the local spatial order-accuracy of the discrete approximation.
∂φ∂x------
ip
∂Nn∂x
----------ip
φnn∑=
∂N∂x-------
∂N∂y-------
∂N∂z-------
∂x∂s------ ∂y
∂s------ ∂z
∂s-----
∂x∂t------ ∂y
∂t------ ∂z
∂t-----
∂x∂u------ ∂y
∂u------ ∂z
∂u------
1–∂N∂s-------
∂N∂t-------
∂N∂u-------
=
P∆nip( )ip
Pip Nn sip tip uip, ,( )Pnn∑=
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Advection Term To complete the discretization of the advection term, the variable must be
approximated in terms of the nodal values of . The advection schemes implemented in
ANSYS CFX can be cast in the form:
(Eqn. 33)
where is the value at the upwind node, and is the vector from the upwind node to
the ip. When using a specified blend, is the average of the adjacent nodal gradients and
when using high resolution scheme, is the nodal gradient of the upwind node.
Particular choices for and yield different schemes.
1st Order Upwind Differencing Scheme
A value of yields a first order Upwind Difference Scheme (UDS). This scheme is very
robust, but it will introduce diffusive discretization errors that tend to smear steep spatial
gradients as shown below:
Numerical Advection Correction Scheme (Specify Blend)
By choosing a value for between 0 and 1, and by setting equal to the average of the
adjacent nodal gradients, the discretization errors associated with the UDS are reduced. The
quantity , called the Numerical Advection Correction, may be viewed as an
anti-diffusive correction applied to the upwind scheme. The choice is formally
second-order-accurate in space, and the resulting discretization will more accurately
φup
φ
φip φup β φ∇ ∆r⋅+=
φup r
∇φ
∇φ
β ∇φ
β 0=
β ∇φ
β∇φ ∆r–
β 1=
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reproduce steep spatial gradients. However, it will introduce dispersive discretization errors
that tend to cause non-physical oscillations in regions of rapid solution variation as shown
below. This scheme is also not as robust as the UDS.
Central Difference Scheme
With this scheme, the value of is set to 1 and is set equal to the local element gradient.
An alternative interpretation is that is evaluated using the trilinear shape functions:
(Eqn. 34)
The resulting scheme is also second-order-accurate, and shares both the desirable and
undesirable attributes of the Numerical Advection Correction Scheme. An additional
undesirable attribute is that this scheme may suffer from serious decoupling issues. While
use of this scheme is not generally recommended, it has proven both useful and accurate
when used with LES based turbulence models - add reference.
High Resolution Scheme
The High Resolution Scheme computes locally to be as close to 1 as possible without
introducing local oscillations, and sets equal to the control volume gradient at the
upwind node. The recipe for is based on the boundedness principles used by Barth and
Jesperson [28]. This scheme is both accurate and bounded since it only reduces to first order
near discontinuities. Note that for vector quantities, such as velocity, is independently
calculated for each vector component.
Compressibility Mass flow terms in the mass conservation equation involve a product of the density, which
depends upon pressure and the advecting velocity. For compressible flows, the
discretization of these terms is made as implicit as possible in each timestep with the use of
the following Newton-Raphson linearization:
(Eqn. 35)
β ∇φ
φip
φip Nn sip tip uip, ,( )φnn∑=
β
∇φ
β
β
ρU( )nA ρnU oA ρoU nA ρoU oA–+≈
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Here, the superscripts k and k-1 respectively indicate the current and previous iterates. This
results in an active linearization involving both the new density and velocity terms for
compressible flows at any Mach number.
MeshDeformation
The integral conservation equations presented above (Eqn. 4) must be modified when the
control volumes deform in time. These modifications follow from the application of the
Leibnitz Rule:
(Eqn. 36)
where is the velocity of the control volume boundary.
As before, the differential conservation equations are integrated over a given control
volume. At this juncture, the Leibnitz Rule is applied, and the integral conservation
equations become:
(Eqn. 37)
(Eqn. 38)
(Eqn. 39)
The transient term accounts for the rate of change of storage in the deforming control
volume, and the advection term accounts for the net advective transport across the control
volume's moving boundaries.
Erroneous sources of conserved quantities will result if the Geometric Conservation Law
(GCL):
(Eqn. 40)
is not satisfied by the discretized transient and advection terms. The GCL simply states that
for each control volume, the rate of change of volume must exactly balance the net volume
swept due to the motion of its boundaries. The GCL is satisfied by using the same volume
recipes for both the control volume and swept volume calculations, rather than by
approximating the swept volumes using the mesh velocities.
tdd φ Vd
V t( )∫ φ∂
t∂------ Vd
V∫ φW j n jd
s∫+=
W j
tdd ρ Vd
V t( )∫ ρ U j W j–( ) n jd
s∫+ 0=
ddt----- ρU i Vd
V t( )∫ ρ U j W j–( )U i n jd
s∫+ P n jd
s∫– µeff
U i∂x j∂
---------U j∂xi∂
----------+⎝ ⎠⎜ ⎟⎛ ⎞
n jds∫ SU i
VdV∫+ +=
ddt----- ρφ Vd
V t( )∫ ρ U j W j–( )φ n jd
s∫+ Γeff
φ∂x j∂
--------⎝ ⎠⎛ ⎞ n jd
s∫ Sφ Vd
V∫+=
tdd Vd
V t( )∫ W j n jd
s∫=
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The Coupled System of Equations
The linear set of equations that arise by applying the Finite Volume Method to all elements
in the domain are discrete conservation equations. The system of equations can be written
in the form:
(Eqn. 41)
where is the solution, b the right hand side, a the coefficients of the equation, i is the
identifying number of the control volume or node in question, and nb means “neighbor”,
but also includes the central coefficient multiplying the solution at the ith location. The
node may have any number of such neighbors, so that the method is equally applicable to
both structured and unstructured meshes. The set of these, for all control volumes
constitutes the whole linear equation system. For a scalar equation (e.g., enthalpy or
turbulent kinetic energy), ainb, and bi are each single numbers. For the coupled, 3D
mass-momentum equation set, they are a (4 x 4) matrix or a (4 x 1) vector, which can be
expressed as:
(Eqn. 42)
and
(Eqn. 43)
(Eqn. 44)
It is at the equation level that the coupling in question is retained and at no point are any of
the rows of the matrix treated any differently (e.g., different solution algorithms for
momentum versus mass). The advantages of such a coupled treatment over a non-coupled
ainbφi
nb
nbi
∑ bi=
φ
φinb
ainb
auu auv auw aup
avu avv avw avp
awu awv aww awp
apu apv apw app i
nb
=
φinb
uvwp
nb
i
=
bi
bu
bv
bw
bp i
=
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or segregated approach are several: robustness, efficiency, generality and simplicity. These
advantages all combine to make the coupled solver an extremely powerful feature of any
CFD code. The principal drawback is the high storage needed for all the coefficients.
Solution Strategy - The Coupled Solver
Segregated solvers employ a solution strategy where the momentum equations are first
solved, using a guessed pressure, and an equation for a pressure correction is obtained.
Because of the ‘guess-and-correct’ nature of the linear system, a large number of iterations
are typically required in addition to the need for judiciously selecting relaxation parameters
for the variables.
ANSYS CFX uses a coupled solver, which solves the hydrodynamic equations (for u, v, w, p)
as a single system. This solution approach uses a fully implicit discretization of the equations
at any given timestep. For steady state problems, the time-step behaves like an ‘acceleration
parameter’, to guide the approximate solutions in a physically based manner to a
steady-state solution. This reduces the number of iterations required for convergence to a
steady state, or to calculate the solution for each timestep in a time dependent analysis.
General Solution
The flow chart shown below illustrates the general field solution process used in the ANSYS
CFX-Solver.
The solution of each set of field equations shown in the flow chart consists of two
numerically intensive operations. For each timestep:
1. Coefficient Generation: The non-linear equations are linearized and assembled into thesolution matrix.
2. Equation Solution: The linear equations are solved using an Algebraic Multigrid method.
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When solving fields in the ANSYS CFX-Solver, the outer- or timestep-iteration is controlled
by the physical time scale or timestep for steady and transient analyses, respectively. Only
one inner (linearization) -iteration is performed per outer-iteration in steady state analyses,
whereas multiple inner-iterations are performed per timestep in transient analyses.
Linear Equation Solution
ANSYS CFX uses a Multigrid (MG) accelerated Incomplete Lower Upper (ILU) factorization
technique for solving the discrete system of linearized equations. It is an iterative solver
whereby the exact solution of the equations is approached during the course of several
iterations.
Initialize Solution Fields and Advance in Time / False Time
START
Solve Mesh Displacement
Solve Wallscale
Solve Hydrodynamic System
Solve Volume Fractions
Solve Additional Variables
Solve Radiation
Solve Energy
Solve Turbulence
Solve Mass Fractions
Solve Full Coupled Particles
Transient?
Yes
No
Coefficient LoopCriteria Satisfied
ConvergenceCriteria/Max
Iteration Satisfied
No
Yes
Advance inFalse Time
Iteration withinthe Timestep
Solve One Way Coupled Particles
STOP
MaximumTime Reached?
No
Yes
Yes
Advance inTime
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The linearized system of discrete equations described above can be written in the general
matrix form:
(Eqn. 45)
where [A] is the coefficient matrix, the solution vector and [b] the right hand side.
The above equation can be solved iteratively by starting with an approximate solution,fn,
that is to be improved by a correction, , to yield a better solution, fn+1, i.e.,
(Eqn. 46)
where is a solution of:
(Eqn. 47)
with rn, the residual, obtained from:
(Eqn. 48)
Repeated application of this algorithm will yield a solution of the desired accuracy.
By themselves, iterative solvers such as ILU tend to rapidly decrease in performance as the
number of computational mesh elements increases. Performance also tends to rapidly
decrease if there are large element aspect ratios present. The performance of the solver can
be greatly improved by employing a technique called ‘multigrid.’
AlgebraicMultigrid
The convergence behavior of many matrix inversion techniques can be greatly enhanced by
the use of a technique called ‘multigrid’. The multigrid process involves carrying out early
iterations on a fine mesh and later iterations on progressively coarser virtual ones. The
results are then transferred back from the coarsest mesh to the original fine mesh.
From a numerical standpoint, the multigrid approach offers a significant advantage. For a
given mesh size, iterative solvers are only efficient at reducing errors which have a
wavelength of the order of the mesh spacing. So, while shorter wavelength errors disappear
quite quickly, errors with longer wavelengths, of the order of the domain size, can take an
extremely long time to disappear. The Multigrid Method bypasses this problem by using a
series of coarse meshes such that longer wavelength errors appear as shorter wavelength
errors relative to the mesh spacing. To prevent the need to mesh the geometry using a
series of different mesh spacings, ANSYS CFX uses Algebraic Multigrid.
Algebraic Multigrid [25] forms a system of discrete equations for a coarse mesh by summing
the fine mesh equations. This results in virtual coarsening of the mesh spacing during the
course of the iterations, and then re-refining the mesh to obtain an accurate solution. This
A φ b=
φ
φ'
φn 1+ φn φ'+=
φ'
Aφ' rn=
rn b Aφn–=
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technique significantly improves the convergence rates. Algebraic Multigrid is less
expensive than other multigrid methods since discretization of the non-linear equations is
only performed once for the finest mesh.
ANSYS CFX uses a particular implementation of Algebraic Multigrid called Additive
Correction [1] This approach is ideally suited to the ANSYS CFX-Solver implementation,
because it takes advantage of the fact that the discrete equations are representative of the
balance of conserved quantities over a control volume. The coarse mesh equations can be
created by merging the original control volumes to create larger ones as shown below. The
diagram shows the merged coarse control volume meshes to be regular, but in general their
shape becomes very irregular. The coarse mesh equations thus impose conservation
requirements over a larger volume and in so doing reduce the error components at longer
wavelengths.
Original mesh
First coarse mesh (virtual)
Next coarse mesh (virtual)
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Residual Normalization Procedure
As described above, the raw residual, [r], is calculated as the imbalance in the linearized
system of discrete equation. The raw residuals are then normalized for the purpose of
solution monitoring and to obtain a convergence criteria. An overview of the normalization
procedure is given below.
For each solution variable, , the normalized residual is given in general by:
(Eqn. 49)
where is the raw residual control volume imbalance, ap is representative of the control
volume coefficient and is a representative range of the variable in the domain. The exact
calculation of ap and is not simple and is not presented here. However, some important
notes are:
1. The normalized residuals are independent of timestep choice.
2. The normalized residuals are independent of the initial guess.
3. For multiphase flows, the volume fraction is considered. This prevents large residuals in
locations where the volume fraction is negligible having a large influence.
Discretization Errors
There are often differences between the exact analytical solution of the modeled differential
equations (see (Eqn. 1)), and the fully converged solution of their discrete representations
(see (Eqn. 4)). These differences are referred to as discretization errors.
Like the principle variables being solved for, errors in these values are both generated by
localized sources and propagated (i.e., amplified, advected, diffused) throughout the
solution domain. Localized sources of error result from the high-order terms that are
excluded from the discrete approximations of terms in the modeled equations. Conversely,
error propagation results from the form of the terms that are included in the discrete
approximations. Both error sources and propagation are affected by the solution and mesh
distributions, as discussed in Controlling Error Sources (p. 296) and Controlling Error
Propagation (p. 297).
Controlling Error Sources
Reducing the source of solution error (i.e., the magnitude of terms excluded in the discrete
approximations) is critical if accurate numerical solutions are desired. The two most
effective strategies of accomplishing this are by increasing the order-accuracy of discrete
approximations (e.g., using the high resolution rather than the upwind difference advection
φ
rφ[ ]rφ[ ]
ap∆φ-------------=
rφ
∆φ
∆φ
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scheme) and/or by reducing the mesh spacing in regions of rapid solution variation. The
former strategy is discussed above (see Advection Term (p. 288)), and implications of the
latter are now considered.
Finely spaced isotropic mesh distributions are ideal, but they are often not tractable. A more
useful strategy for reducing sources of solution error is to generate anisotropic meshes with
fine spacing in directions of most rapid solution variation and relatively coarse spacing in
other directions. This is exemplified by typical boundary layer meshes that are compressed
in the direction of most rapid solution variation (i.e., normal to the wall).
It is important to realize, however, that sources of solution error are also affected by poor
geometrical mesh quality (see Measures of Mesh Quality (p. 371 in "ANSYS CFX-Solver
Modeling Guide")). In particular, the error source contributions due to the discretization of
transient/storage, diffusion, source and Rhie-Chow redistribution terms increase with mesh
anisotropy. This is why, for example, high orthogonality and low expansion factors are
recommended in boundary layer meshes where diffusive transport dominates.
Controlling Error Propagation
Controlling the transport and, more importantly, the amplification of error, is also critical,
given that sources of solution error can often only be reduced and not eliminated. Little can
be done to reduce error transport because error is subject to the same physical processes
(e.g., advection, diffusion) as the conserved quantities. Fortunately, the amplification of
error is more easily controlled.
Errors are amplified by strong unphysical influences in the discrete form of the modeled
equations. These unphysical influences will lead to convergence difficulties, and in extreme
cases, complete divergence. Similar to error sources, error amplification is controlled
through the choice of discretization and/or the mesh distribution.
As highlighted above, (unphysical) negative influences may be introduced into the
discretization through the finite-element shape functions, and these influences may grow
as geometrical mesh quality (see Measures of Mesh Quality (p. 371 in "ANSYS CFX-Solver
Modeling Guide")) deteriorates. Tri-linear shape functions, for example, are more
susceptible to negative influences than linear-linear shape functions. Nevertheless, tri-linear
shape functions are used as much as possible due to their improved accuracy, and
linear-linear shape functions are used whenever solution robustness is critical.
Geometrical mesh quality is important regardless of the shape functions used. An excellent
example of this occurs when mesh elements are folded. If the mesh is sufficiently folded,
control volumes become negative and the balance between the transient/storage and flow
terms is rendered physically invalid. The amount of a conserved quantity within a control
volume will actually decrease given a net flow into the control volume.
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Page iANSYS CFX-Solver Theory Guide
Index
A
adaptioncriteria 61to geometry 63
additional variablesscalar transport equation 40
Schmidt number 41wall function model 113
adiabatic boundary conditions 57advection scheme 288Advection-diffusion equation 65algebraic multigrid 294Analysis
proximate analysis of coal 200Antoine equation 197Area porosity 65
B
baseline (BSL) k-omega model 79black body
energy spectrum 265blending functions 82Boltzmann’s constant 265Boundary Conditions
DES 103boundary conditions
adiabatic 57mathematical models 46
inlet (subsonic) 46inlet (supersonic) 51opening 55outlet (subsonic) 51outlet (supersonic) 54symmetry plane 58wall 56
Boussinesq model 34
buoyancy reference temperature 34thermal expansivity 34
buoyancy 33Boussinesq model 34full buoyancy model 34
C
CFX-5definitions 6
Char oxidation 201chemistry post-processing 258CHT
conjugate heat transfer 33Coal combustion 198
char oxidation 201decomposition 198devolatilization 199gas phase 198radiative pre-heating 203
coefficient of thermal expansion 4combustion
combined eddy dissipation/finite ratechemistry model 232
finite rate chemistry model 230laminar flamelet model 233NOx model 252phasic 251premixed and partially premixed
(Zimont) model 239soot model 259
conducting solids 267conjugate heat transfer
CHT 33continuity equation 23continuity source 45
Index: D
Page ii ANSYS CFX-Solver Theory Guide
D
Darcy’s law 66density
effective density 124fluid density 124
Devolatilization 199dimensions 2discrete transfer 270discretisation 277domain
temperature 9drag, coefficient 138
E
eddy viscosityturbulence models 72
energyequation 23
multiphase flow, multifluidmodel 147
equation of state 25ideal gas 25
molecular weight 25
F
faceelement 278
finite rate chemistry model 230finite volume method 277free surface
flow 184full buoyancy model 34
G
GGI 119governing equations
continuity equation 23energy equation 23equation of state 25momentum equation 23multiphase flow 129
H
Heat transferparticle transport 196
heat transfercoefficient 149, 167wall function model 112
hierarchical refinement 62high resolution advection scheme 289homogeneous
hydrodynamic equations 130homogeneous multiphase 127
I
identity matrix 5incremental adaption 62inflation
and mesh adaption 63inlet (subsonic)
mathematical model 46inlet (supersonic)
mathematical model 51
K
k-epsilon turbulence model 75k-omega models
baseline(BSL) 79shear stress transport (SST) 81
k-omega turbulence model 78Kronecker delta function 5
L
linear equation solution 293
M
mass source 45multiphase 131
Mass transferparticle transport 197
mathematical formulation, turbulencemodels 107mathematical notation 20mesh adaption 61
Index: N
ANSYS CFX-Solver Theory Guide Page iii
adapting to the geometry 63advice 64
discontinuities 64and inflation 63implementation 62limitations 64
mesh refinementimplementation 62
MFR 119modelling
flow at the walladditional variables 113heat transfer 112
flow near the wall 107modified pressure 6momentum equation 23
multiphase flow, multifluid model 128momentum sources 43Monte Carlo 271multicomponent flow
componentsalgebraic equation 37constraint equation 37transport equation 35
fluid properties 38multiphase 131
multigrid, algebraic 294multiphase
free surface 184multiphase flow 123
inter-phase drag 132inter-phase drag models
Schiller-Naumann drag model 134inter-phase transfer 132mixture model 132
interfacial area equation 126Prandtl number 127Reynolds number 127
multicomponent multiphase flow 131particle model 132
interfacial area equation 125transport equations, multifluid
model 128turbulence 163
N
Navier-Stokes equations 22normalised residuals 296NOx model 252
numerical advection correctionscheme 288numerical discretisation 277Nusselt number 149, 196
O
openingmathematical model 55
outlet (subsonic)mathematical model 51
outlet (supersonic)mathematical model 54
P
p-1 radiation model 269particle Reynolds number 126Particle transport
heat transfer 196liquid evaporation
liquid evaporation model 197mass transfer 197
phasic combustion 251Planck’s constant 265Porosity
area 65volume 65
Porous regionsarea porosity, definition 65equations 65resistance, definition 65volume porosity, definition 65
Prandtl numbermultiphase 126
pressure-strain terms, Reynolds stressmodel 87Proximate analysis 200
R
Radiationradiative pre-heating 203
radiationRosseland model 268theory 263through conducting solids 267
RANS equations 70
Index: S
Page iv ANSYS CFX-Solver Theory Guide
reference pressure 6Resistance 66Reynolds Averaged Navier Stokes (RANS)equations 70Reynolds number
particle 126Reynolds stress model 86Reynolds stress turbulence model 85RNG k-epsilon model 77rotating frame quantities 17
S
scalable wall-functions 108Scalar
advection-diffusion equation 65scalar dynamic diffusivity 5shape functions 283shear strain rate 16shear stress transport (SST) k-omega basedmodel 81solid domain 33solver
yplus and yplus 109soot model 259sources 43
momentum 43specify blend 288SST k-omega based model 81static pressure 6static temperature 9statistical turbulence models 70subdomain
sources 43surface tension
in free surface flows 184symbols
list of 2symmetry plane
mathematical model 58
T
totalenthalpy 23
total pressure 14, 124total temperature 10transport equations 23turbulence 69
closure models 70turbulence model
k-epsilon turbulence model 75k-omega turbulence model 78Reynolds stress 85RNG k-epsilon turbulence model 77zero equation 74
turbulence models 69two equation turbulence models 75
U
upwind difference scheme (UDS) 288
V
vectoroperators 20
volumefraction 124
Volume porosity 65Von Karman constant 5
W
wallmathematical model 56
wall scale 82Wilcox k-omega model 78
Z
zero equation turbulence model 74