ANSYS CFX Solver Theory

ANSYS CFX-SolverTheory Guide

ANSYS CFX Release 11.0

December 2006

ANSYS, Inc.

Southpointe

275 Technology Drive

Canonsburg, PA 15317

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http://www.ansys.com

(T) 724-746-3304

(F) 724-514-9494

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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

Page vi ANSYS CFX-Solver Theory Guide

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

ANSYS CFX-Solver Theory Guide Page vii

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

Table of Contents: Particle Transport Theory

Page viii ANSYS CFX-Solver Theory Guide

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

The Algebraic Slip Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Phasic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Bulk Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Drift and Slip Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Derivation of 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

Table of Contents: Combustion Theory

ANSYS CFX-Solver Theory Guide Page ix

Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Simple Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Liquid Evaporation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Oil Evaporation/Combustion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

Page x ANSYS CFX-Solver Theory Guide

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

Page 1ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved.Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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 masstransfer in ANSYS CFX for single-phase, single and multi-component flow withoutcombustion or radiation. It is designed to be a reference for those users who desire a moredetailed understanding of the mathematics underpinning the ANSYS CFX-Solver, and istherefore not essential reading. It is not an exhaustive text on CFD mathematics; a referencesection 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

Page 2 ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved.Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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, dimensionsand where applicable, values. Dimensionless quantities are denoted by 1. The values ofphysical constants (or their default values) are also given.

More information on the notation used in the multiphase and multicomponent chapters isavailable.

• Multiphase Notation (p. 124)

• Multicomponent Notation (p. 35).

L M T ! A

Symbol Description Dimensions Valuelinear energy source coefficient

Stanton number

linear resistance coefficient

quadratic resistance coefficient

- turbulence model constant

RNG - turbulence model coefficient


RNG - turbulence model constant


Reynolds Stress model constant


fluid speed of sound

CE M L 1– T 3– ! 1–

Ch 1

CR1 M L 3– T 1–

CR2 M L 4–

C"1 k " 1 1.44

C"1RNG k " 1 1.42 f h–

C"2 k " 1 1.92

C"2RNG k " 1 1.68

Cµ k " 1 0.09

CµRS 1

CµRNG k " 1 0.085

c L T 1–


ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. © 1996-2006 ANSYS Europe, Ltd. All rights reserved. Page 3Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.

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


specific heat capacity at constantpressure

specific heat capacity at constantvolume



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– ! 1–

cv L2 T 2– ! 1–

c"1 1 1.45

c"2 1 1.9

DAB L2 T 1–

D#$# %⁄ L2 T 1–

d LE 1 9.793f µ 1 0.01

f h k " 1

g L T 2–

h hstat, L2 T 2–

hc M T 3– ! 1–

htot L2 T 2–

k L2 T 2–

M U c⁄ 1m M T 1–

Pk M L 1– T 3–

p pstat, M L 1– T 2–

pref M L 1– T 2–

ptot M L 1– T 2–

p' M L 1– T 2–

R0 L2 T 2– ! 1– 8314.5



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)


diffusivity

molecular diffusion coefficient of

component


Re rU d m⁄ 1r Lr& & 1

SE M L 1– T 3–

SM M L 2– T 2–

SMS M L 3– T 1–

Sct µt/$t 1

s&' &'

M T 1–

T T stat, !

Tdom !

T ref !

T sat !

T tot !

U Ux y z, , L T 1–

U L T 1–

u L T 1–

W f M L 1– T 3–

w 1

Y A

&&

''

' ! 1–

'RNG k " 1 0.012

$ M L 1– T 1–

$AA M L 1– T 1–



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






shear stress or sub-grid scale stressmolecular stress tensor

specific volume

additional variable (non-reacting scalar)

general scalar variable

angular velocity


$# M L 1– T 1–

$t M L 1– T 1–

( 1

" L2 T 3–

) M L 1– T 1–

* 1 0.41+ M L T 3– ! 1–

µ M L 1– T 1–

µt M L 1– T 1–

µeff µ µt+ M L 1– T 1–

% M L 3–

Pr cpµ +⁄ 1

Prt cpµt +t⁄ 1

,k k 1 1.0

," k " 1 1.3

,- k - 1 2

,"RS 1

,kRNG k " 1 0.7179

,"RNG k " 1 0.7179

. M L 1– T 2–

/ M 1– L3

# M L 3–

0- T 1–

A B C AB C

& ' 1 & ' 1



Such quantities are only used in the chapters describing multicomponent and multiphaseflows.

• Multicomponent Flow (p. 35)

• Multiphase Flow Theory (p. 123)

Variable Definitions

IsothermalCompressibility

The isothermal compressibility defines the rate of change of the system volume withpressure. For details, see Variables Relevant for Compressible Flow (p. 62 in "ANSYSCFX-Solver Manager User's Guide").

(Eqn. 1)

IsentropicCompressibility

Isentropic compressibility is the extent to which a material reduces its volume when it issubjected to compressive stresses at a constant value of entropy. For details, see VariablesRelevant 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 otherpressure values are taken. All relative pressure specifications in ANSYS CFX are relative to theReference Pressure. For details, see Setting a Reference Pressure (p. 10 in "ANSYSCFX-Solver Modeling Guide").

(Eqn. 3)

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

(Eqn. 6)

1%---–2 3

4 5 d%dp------2 3

4 5T

1%---–2 3

4 5 d%dp------2 3

4 5S

Pref

pstat

pabs pstat pref+=

k "k

p' pstat2%k

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



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 staticenthalpy. General changes in enthalpy are also used by the solver to calculatethermodynamic properties such as temperature. To compute these quantities, you need toknow how enthalpy varies with changes in both temperature and pressure. These changesare 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 isnot included. For example, the second term is zero for materials which use the Ideal Gasequation of state or materials in a solid thermodynamic state. In addition, the second termis also dropped for liquids or gases with constant specific heat when you run the thermalenergy 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 ustatpstat%stat---------+=

dh h6T6------2 3

4 5pdT h6

p6------2 34 5

Tdp+=

dh cpdT 1%--- 1 T

%---%6T6------2 3

4 5p

+ dp+=

cp %

h T p,( )% cp

p1 T1,( ) p2 T2,( )

hd

h2 h1– cpT1

T2

7 dT 1%--- 1 T

%---%6T6------2 3

4 5p

+p1

p2

7 dp+=



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 numericallyevaluated from the using a two point central difference formula. In addition, the ANSYSCFX-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 theenthalpy and entropy derivatives.

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

(Eqn. 11)

%cp

T 2

2

66 h2 38 94 5

p p2

2

66 h2 38 94 5

T=

T 2

2

66 cp

2 38 94 5

p

1%---

%6T6------2 3

4 5p

1%---

2T%2-------

%6T6------2 3

4 5p

+2 34 5 T

%2----- T 2

2

66 %2 38 94 5

p+=

T

(p2,T2)(p1,T1)

p

h

21

htot hstat12-- U U:( )+=



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. Inrotating 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 ModelingGuide").

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 orthermal energy equations).

The static temperature is calculated using static enthalpy and the constitutive relationshipfor the material under consideration. The constitutive relation simply tells us how enthalpyvaries 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 forenthalpy:

(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 oftemperature 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 byinverting the table.

Material with Variable Density and Specific HeatTo properly handle materials with an equation of state and specific heat that vary asfunctions of temperature and pressure, the ANSYS CFX-Solver needs to know enthalpy as a

function of temperature and pressure, .

U

Tdom

T stat

% cp

hstat href– cp T stat T ref–( )=

T ref 0 K[ ]= href 0 J kg( )⁄[ ]=

hstat href– cp T( ) TdT ref

T stat

7=

cp

h T p,( )



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 byinverting 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 exactlythe same way as static temperature, except that total enthalpy is used in the propertyrelationships.

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 kineticenergy. This can be illustrated by starting with the total enthalpy form of the constitutiverelation:

(Eqn. 15)

and substituting expressions for Static Enthalpy and Total Pressure for an incompressiblefluid:

(Eqn. 16)

(Eqn. 17)

some rearrangement gives the result that:

(Eqn. 18)

for this case.

h T p,( )

h T p,( )

h T p,( )

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

hstat pstat T stat

% cp

htot href– cp T tot T ref–( )=

htot hstat12-- U U:( )+=

ptot pstat12--% U U:( )+=

T tot T stat=



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 temperatureexcept 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 "– kcp

T tot T statU U:2cp------------- k

cp-----+ +=

htot href– cp T( ) TdT ref

T tot

7=

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

htot T tot ptot

Tds dh dp%------–=



which can be rearranged to give:

(Eqn. 25)

Depending on the equation of state and the constitutive relationship for the material, youcan arrive at various forms for calculating the changes in entropy as a function oftemperature 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 changesin 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%T-------–=

dh cpdT dp%------+=

ds cpdTT-------=

s sref– cpT

T ref---------2 3

4 5log=

s sref–cp T( )

T-------------- TdT ref

T7 R p

pref--------2 3

4 5log–=

cp

cp

s sref– cpT

T ref---------2 3

4 5log R ppref--------2 3

4 5log–=



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 ANSYSCFX-Solver computes a table of values at a number of temperature and pressure points. Thepoints 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 andpressure. Expressions for the derivatives are found by substituting the formula for generalenthalpy changes into the second law to get the following expression for changes inentropy 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 thederivative with respect to pressure must be computed by numerically differentiating theequation of state. Note that when properties are specified as a function of temperature andpressure using CEL expressions the differential terms in Equation 31 must also satisfy theexact differential relationship:

See the previous section on Static Enthalpy for more details.

s T p,( )

dscpT-----dT 1

%2----- %6

T6------2 34 5

pdp+=

ds s6T6------2 3

4 5pdT s6

p6------2 34 5

Tdp+=

s6T6------2 3

4 5p

cpT-----=

s6p6------2 3

4 5T

1%2-----

%6T6------2 3

4 5p

=

T 2

2

66 s2 38 94 5

p p2

2

66 s2 38 94 5

T=



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 andpressure, 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 topressure without losses. The following three sections describe how total pressure is

%cp

s2 s1–cpT-----dT

T1

T2

7 1%2-----

%6T6------2 3

4 5

p1

p2

7 pdp+=

s h p,( )

ds s6h6------2 3

4 5pdh s6

p6------2 34 5

hdp+=

ds dhT------

dp%T-------–=

s6h6------2 3

4 5p

1T---=

s6p6------2 3

4 5h

1%T-------–=

h ph T p,( ) T h p

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

s sref– 1T--- hd

href

h

7 1%T-------

pref

p

7 dp–=

h p

ptot



computed for a pure component material with constant density, ideal gas equation of stateand a general equation of state (CEL expression or RGP table). For details, see MultiphaseTotal Pressure (p. 124).

incompressible FluidsFor incompressible flows such as those of liquids and low speed gas flows, the total pressureis 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 lawof 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 righthand side by the equation of state. For an ideal gas, the constitutive relation and equationof 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--% U U:( )+=

dh dp%------=

dh cp T( )dT=

% pRT-------=

ptot pstat1R---

cp T( )T-------------- Td

T stat

T tot

72 34 5exp=

T stat T tot

cp cp



Material with Variable Density and Specific HeatTotal pressure calculations in this case are somewhat more involved. You need to calculatetotal pressure given the static enthalpy and static pressure, which the ANSYS CFX-Solversolves for. You also want to assume that the local state variations, within a control volumeor 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 strainrate:

(Eqn. 47)

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

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

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

s h p,( )

s h p,( )

htot pstat

hstat htot

sstat s hstat pstat,( )=

stot sstat=

stot s htot ptot,( )=

Sij12--

U i6x j6---------

U j6xi6---------+2 3

4 5=

sstrnr 2U i6x j6---------Sij

12--

=



with velocity components , , , this expands to:

(Eqn. 48)

The viscosity of non-Newtonian fluids is often expressed as a function of this scalar shearstrain 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 stationaryframe total pressure are computed in the same manner as in Total Pressure. First you startwith the relative total enthalpy, rothalpy and stationary frame total enthalpy:

(Eqn. 52)

(Eqn. 53)

(Eqn. 54)

U x U y U z

sstrnr 2U x6x6----------2 3

4 52 U y6

y6----------2 34 5

2 U z6z6---------2 3

4 52

+ +; <= >? @

=

U x6y6----------

U y6x6----------+2 3

4 5 2 U x6z6----------

U z6x6---------+2 3

4 5 2 U y6z6----------

U z6y6---------+2 3

4 5 2+ + +

12--

Urel Ustn - RA–=

- R Ustn

ptot pstat12--% Urel Urel: - RA - RA:( )–( )+=

pstat

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

htot hstat12-- Urel Urel:( )+=

I hstat12-- Urel Urel: - RA - RA:( )–( )+=

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



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 youcan compute relative total temperature, total temperature and stationary frame totaltemperature 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 thesection describing total pressure. For details, see Total Pressure (p. 14).

The names of the various total enthalpies, temperatures, and pressures when visualizingresults 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

7=

I href– cp T( ) TdT ref

T tot

7=

htot,stn href– cp T( ) TdT ref

T tot,stn

7=

cp

T tot,rel T statUrel Urel:

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

T tot T statUrel Urel: - RA - RA:( )–( )

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

T tot,stn T statUstn Ustn:

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

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



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 wayas 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 aone-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 thevelocity and length scale are based on the mass flow into the control volume and thedimension 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 uBtBx---------=

u tB xB



For explicit CFD methods, the timestep must be chosen such that the Courant number issufficiently small. The details depend on the particular scheme, but it is usually of orderunity. As an implicit code, ANSYS CFX does not require the Courant number to be small forstability. However, for some transient calculations (e.g., LES), one may need the Courantnumber 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-Solverdocumentation.

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 kC

C x66

y66

z66, ,=

0 x y z, ,( ) 0

0C 06x6------i

06y6------ j

06z6------k+ +=

U x y z, ,( )

UUx

U y

Uz

=

U

UC•U x6x6----------

U y6y6----------

U z6z6---------+ +=



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 becombined 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 mayhelp you if you are used to index notation.

U V

U VDU 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

=

%U UD( )C•

x66 %UxUx( ) y6

6 %U yUx( ) z66 %UzUx( )+ +

x66 %UxU y( ) y6

6 %U yU y( ) z66 %UzU y( )+ +

x66 %UxUz( ) y6

6 %U yUz( ) z66 %UzUz( )+ +

=

T

0C

06x6------

06y6------

06z6------

=

0CT 06

x6------06y6------

06z6------

=

(1 0 00 1 00 0 1

=

Basic Solver Capability Theory: Governing Equations


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 arederivatives 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 theirconservation form.

If you are new to CFD, review the introduction. For details, see Computational FluidDynamics (p. 1 in "ANSYS CFX Introduction").

A list of recommended books on CFD and related subjects is available. For details, seeFurther Background Reading (p. 6 in "ANSYS CFX Introduction").

For all the following equations, static (thermodynamic) quantities are given unlessotherwise stated.

UC•U i6xi6---------=

i

U VD U iV j U V

U iV jk U V

%U UD( )C•

xi66 %U iU j( )

%U UD( )C•

xi66 %U iU j( )

x j66 %U iU j( )

a b• a b aijbij



Transport Equations

In this section, the instantaneous equation of mass, momentum, and energy conservationare presented. For turbulent flows, the instantaneous equations are averaged leading toadditional terms. These terms, together with models for them, are discussed in Turbulenceand Wall Function Theory (p. 69).

The instantaneous equations of mass, momentum and energy conservation can be writtenas 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)

%6t6------ %U( )C•+ 0=

%U( )6t6---------------- %U UD( ) Cp C .•+– SM+=C•+

.

. µ CU CU( )T 23--(C U•–+2 3

4 5=

%htot( )6t6

------------------- p6t6------ %Uhtot( )C•+– +CT( ) C U .•( )• U SM S+•+ E+C•=

htot h T p,( )

htot h 12--U

2+=

C U .•( )•

U SM•

K

K 12--U

2=

U

%K( )6t6

--------------- %UK( ) U– Cp• U C .•( )• U SM•+ +=C•+



Subtracting this equation from the total energy equation (Eqn. 80) yields the thermal energyequation:

(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 internalenergy. For this reason, the thermal energy equation should not be used in thissituation, particularly for subcooled liquids.

Equations of State

In ANSYS CFX, the flow solver calculates pressure and static enthalpy. Finding densityrequires that you select the thermal equation of state and finding temperature requires thatyou select the constitutive relation. The selection of these two relationships is notnecessarily independent and is also a modeling choice.

The thermal equation of state is described as a function of both temperature and pressure:

(Eqn. 87)

%h( )6t6

-------------- p6t6------ %Uh( ) +CT( ) U Cp .:CU+• SE++C•=C•+–

.:CUh e

h e P%---+=

%e( )6t6

------------- %Ue( ) +T( ) pC U .:CU+• SE++C•=C•+

pC U•

e

P %⁄

P %⁄

% % p T,( )=



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 abuilt-in option for simulating real gases. It is also available through several pre-suppliedCFX-TASCflow RGP files. The Vukalovich Virial equation of state is also available but currentlyonly by using CFX-TASCflow RGP tables.

Redlich Kwong Gas PropertiesThe Redlich-Kwong equation of state was first published in 1949 and is considered one ofthe most accurate two parameter corresponding states equations of state. This equation ofstate is quite useful from an engineering standpoint because it only requires that the userknow the fluid critical temperature and pressure. More recently, Aungier (1995) [96] hasmodified the Redlich-Kwong equation of state so that it provides much better accuracy nearthe critical point. The Aungier form of this equation of state is used by ANSYS CFX and isgiven by:

(Eqn. 91)

cp

cp cp p T,( )=

cp

cp cp T( )=

% cp

%w p pref+( )

R0T---------------------------=

w R0

p RTE b– c+-------------------- a T( )

E E b+( )--------------------–=



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 istabulated 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 thecritical 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 oftemperature.

In order to provide a full description of the gas properties, the flow solver must also calculateenthalpy and entropy. These are evaluated using slight variations on the generalrelationships for enthalpy and entropy which were presented in the previous section onvariable definitions. The variations depend on the zero pressure, ideal gas, specific heat

ca T( )

a b c

a a0TTc------2 34 5 n–

=

a00.42747R2Tc

2

pc---------------------------------=

b0.08664RTc

pc-----------------------------=

cRTc

pca0

Ec Ec b+( )-------------------------+------------------------------------- b Ec–+=

a T( ) n 0.5=n n

n-

n 0.4986 1.2735- 0.4754-2+ +=

- log– 10pvpc-----2 34 5 1–=

pv T 0.7Tc=



capacity and derivatives of the equation of state. The zero pressure specific heat capacitymust 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 toinfinite 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 energychange along this path is:

(Eqn. 99)

Once the internal energy is known, then enthalpy is evaluated from internal energy:

T vT ref vref

T v

ud cv Td T pdTd------2 3

4 5v

p–2 34 5 vd+=

cv

u T v,( ) u T ref vref,( )– =

T pdTd------2 3

4 5v

p–2 34 5 vT ref

dvref

F

7 cv0 TdT ref

T

7 T pdTd------2 3

4 5v

p–2 34 5 vTd

v

F

7–+

h u pE+=



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. Theselection of this state is arbitrary, and can be set by the user, but by default, ANSYS CFX usesthe normal boiling temperature (which is provided) as the reference temperature and thereference pressure is set to the value of the vapor pressure evaluated using Equation 106 atthe normal boiling point. The reference enthalpy and entropy are set to zero at this point bydefault, but can also be overridden if desired.

Other properties, such as the specific heat capacity at constant volume, can be evaluatedfrom 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------2 3

4 5v

vT refd

vref

F

7cv0T------ Td

T ref

T

7 R ppref--------2 34 5ln– pd

Td------2 34 5

vvTd

v

F

7–+

cpo

cEu6T6------2 3

4 5E

u06T6--------

n n 1+( )abT------------------------ log 1 b

v--+2 34 5–= =

u0

u0 u uref– cpo T( ) R–( )T ref

T

7 dT= =

cp cv

cp cv ET'2

*-----+=



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 phasechange or Eulerian thermal phase change model) or dry calculations, the flow solver needsto know the form of the vapor pressure curve. Pressure and temperature are notindependent in the saturation dome so vapor saturation properties are evaluated by firstassuming an equation which gives the dependence of vapor pressure on temperature, andthen substituting that into the equation of state.

For materials that use the Redlich Kwong equation of state ANSYS CFX approximates thevapor pressure curve using a form given by Aungier (2000) [65]:

(Eqn. 106)

Vapor saturation properties are calculated by evaluating the equation of state andconstitutive 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 thecompressed liquid regime.

Instead, when you select to use the Redlich Kwong equation of state for a liquid, theproperties are assumed to vary along the vapor pressure curve as a function of saturationtemperature. These properties are approximate and should only be used when the amountof liquid in your calculation will be small. For example, they work well with the equilibriumcondensation model or non-equilibrium small droplet phase change model.

To derive the liquid enthalpy and entropy, such that they are completely consistent with thegas 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.

' *

'

p6T6------2 3

4 5E

E p6E6------2 3

4 5T

------------------–=

* 1

E p6E6------2 3

4 5T

------------------–=

log10pvpc-----2 34 5 7

3-- 1 -+( ) 1TcT------–2 3

4 5=



To calculate saturated liquid densities, an alternative equation of state originally publishedby Yamada and Gunn (1973), is used by ANSYS CFX that gives liquid specific volume as afunction of temperature:

(Eqn. 107)

This equation is convenient because it only requires knowledge of the critical volume andtemperature 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 andthe 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 saturationentropy:

(Eqn. 110)

Prediction of liquid specific heat capacity with the Redlich Kwong equation has a similarproblem 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)

E Ec 0.29056 0.08775-–( ) 1 T T c⁄–( )2 7⁄

=

Tc T Tc

hf,s hg,s hfg–=

hfg

hfgRTc---------- 7.08 1 T

Tc------–2 3

4 5 0.35410.95- 1 T

Tc------–2 3

4 5 0.456+=

sf sghfgT------–=

cp cp0 0.5 2.2-+( ) 3.67 11.64 1 TTc------–2 3

4 5 40.634 1 T

Tc------–2 3

4 5 1–+ +2 3

4 5+=

cp cv=

p %RT B1 B2% B3%2 B4%

3+ + +( )=



This equation of state is known to be valid over pressure and temperature ranges of 0.01 barto 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 steamproperties. The database is fully described elsewhere [125], but a summary will be providedin this section. The IAPWS database uses formulations for five distinct thermodynamicregions 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 thermodynamicspace at very high temperatures (1073.15 - 2273.15 K) and reasonably low pressures (0-10MPa) that can be adequately described using other property databases already in ANSYSCFX (i.e., Ideal Gas EOS with NASA specific heat and enthalpy). Furthermore, since this regionis not defined for pressures up to 100 MPa, as is the case for regions 1, 2 and 3, problemsarise in filling out the pressure-temperature space in the tables when temperatures exceed1073.15 K and pressures exceed 10 MPa. The database implemented in ANSYS CFX thereforecovers temperatures ranging from 273.15 to 1073.15 K and pressures ranging from 611 Pato 100 MPa.

The reference state for the IAPWS library is the triple point of water. Internal energy, entropyand 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



In ANSYS CFX, the analytical equation of state is used to transfer properties into tabularform, which can be evaluated efficiently in a CFD calculation. These IAPWS tables aredefined in terms of pressure and temperature, which are then inverted to evaluate states interms of other property combinations (such as pressure/enthalpy or entropy/enthalpy).When developing the IAPWS database for ANSYS CFX, therefore, properties must beevaluated as functions of pressure and temperature. For the most part, this involves astraightforward 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 Region3 (near the critical point), where the EOS is defined explicitly in terms of density andtemperature. 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 subcooledliquid and superheated vapor side leading up to critical conditions. Therefore, dependingon the pressure-temperature state, one may be evaluating a subcooled liquid or asuperheated vapor with the same EOS. To apply the Newton-Raphson scheme in a reliableway, one must detect on which side of the saturation dome the pressure-temperature stateapplies, and apply an appropriate initial guess. Such an iteration scheme, including logic foran initial guess, has been implemented in CFX so that table generation around the criticalregion is possible.

Metastable Superheated Liquid/Supercooled Vapor StatesThe IAPWS library also extends to metastable states, so that this equation of state is availablefor nonequilibrium phase change models such as the Droplet Condensation Model. The EOSfor regions 1 and 3 in Figure 1 are stated to have reasonable accuracy for metastable statesclose to the saturation line d liquid states [125]. However, the term “reasonable” is notquantified, and therefore the degree to which the extrapolation of the EOS can be appliedis unknown.

In region 2, an additional set of equations have been developed for supercooled vaporconditions 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 intosupercooled conditions, but it does not match smoothly with the specialized supercooledequations below 10 MPa.

Numerical Testing to Delineate Metastable RegionsIn order to make the IAPWS database as robust as possible, numerical testing has been doneto determine approximate metastable vapor/liquid spinodal lines. Figure 2 is given todemonstrate these spinodal lines which are essentially boundaries up to which metastableconditions can exist. These would be defined similar to saturation curves as functions of

Basic Solver Capability Theory: Buoyancy


either temperature or pressure. The IAPWS database tables are always generated up tothese limits regardless of what flow models are specified (i.e., equilibrium ornon-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 aresolved, but with no flow. This is known as conjugate heat transfer, and the solid regions areknown as solid domains.

Within solid domains, the conservation of energy equation is simplified since there is noflow inside a solid, thus conduction is the only mode of heat transfer. The heat conductionthrough 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)

t66 %cpT( ) C +CT( ) SE+•=

% cp +

SM,buoy % %ref–( )g=

Basic Solver Capability Theory: Buoyancy


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 (CartesianCoordinates). For details, see Buoyancy: Option (p. 148 in "ANSYS CFX-Pre User's Guide"). Ifunset (option set to Automatic), the solver defaults it to the centroid of a pressure-specifiedboundary (if one exists), or to the pressure reference location (if no pressure-specifiedboundary exists).

Absolute pressure is used to evaluate fluid properties which are functions of pressure, andcan 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, multiphaseflow, or a fluid having density set as a function of pressure, temperature, or other fieldvariables.

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 isapproximated as:

(Eqn. 116)

where is the thermal expansivity:

(Eqn. 117)

and is the buoyancy reference temperature.

% %ref–

%ref

pabs p pref %ref g r rref–( )+ +=

rref

% %ref–

%ref

% %ref– %ref' T T ref–( )–=

'

' 1%---6%6T------- p

–=

T ref

Basic Solver Capability Theory: Multicomponent Flow


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 thefluid are transported within the fluid.

The bulk motion of the fluid is modeled using single velocity, pressure, temperature andturbulence fields. The influence of the multiple components is felt only through propertyvariation by virtue of differing properties for the various components. Each component hasits' own equation for conservation of mass. After Reynolds-averaging (see TurbulenceModels (G. 69 HE IJKLML N#OPLQ+R"% ST"Q%U $/H("I)) this equation can beexpressed 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 CA B C A B

C % B %B

6%i6t--------

6 %iU j( )6x j

-------------------+

66x j-------- %i U ij U j–( ) %i'' U j ''–( )– Si+=

%i i

U j %iU ij( ) %⁄V=U ij i



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 thestandard 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 concentrationgradients, a pressure gradient, external forces or a temperature gradient. Of these possiblesources of relative motion among the mixture components, the primary effect is that ofconcentration 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. Fordetails, see Fluid Models Tab (p. 149 in "ANSYS CFX-Pre User's Guide"). A detaileddescription of the effects of the relative mass flux term and various models for it may befound 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)

%i U ij U j–( )

Si i

6%6t------

6 %U j( )6x j

------------------+ 0=

Si

%i U ij U j–( )$i%-----6%i6x j--------–=

$i %Di Di

i

Y i%i%----=

6 %Y i( )6t-----------------

6 %U jY i( )6x j

------------------------+

66x j-------- $i

6Y i6x j--------2 3

4 5 66x j-------- %Y i

WWU jWW( )– Si+=



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 theequations 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 thesolution 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 knownas the constraint component because its mass fraction is determined by the constraintequation:

(Eqn. 126)

The performance of the ANSYS CFX-Solver will not be affected by your choice of constraintcomponent.

%Y iWWU j

WW µtSct------

6Y i6x j--------=

Sct

Y i

6 %Y i( )6t-----------------

6 %U jY i( )6x j

------------------------+ 66x j-------- $ieff

6Y i6x j--------2 3

4 5 Si+=

$ieff$i

µtSct-------+=

Y i

NC 1–

Y ii A B C …, , ,=

N C

V 1=



Multicomponent Fluid Properties

The physical properties of general multicomponent mixtures are difficult to specify. Thedefault treatment in ANSYS CFX, Release 11.0 makes the assumption that the componentsform an ideal mixture. For details, see Ideal Mixture (p. 15 in "ANSYS CFX-Solver ModelingGuide").

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 thecomponent, 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 mixturetemperature 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

%i Mi V⁄= i

V i

%iX Y Mi V i⁄=

V

1V iV-----

i A B…,=

N C

VMi %iX Y⁄

Mi %i⁄---------------------i A B…,=

N C

V= =

%i%iX Y----------

i A B…,=

N C

V%Y i%iX Y----------

i A B…,=

N C

V= =

1%---

Y i%iX Y----------

i A B…,=

N C

V=

Y i

%i %iX Y %i

%iX Y

& Y i &ii A B…,=

N C

V=



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 singlecomponent fluid. Extending this equation for multicomponent fluids involves adding anadditional diffusion term to the energy equation:

(Eqn. 130)

For turbulent flow, this term is Reynolds-averaged (see Turbulence Models (G. 69 HEIJKLML N#OPLQ+R"% ST"Q%U $/H("I)) giving:

(Eqn. 131)

This expression introduces several terms involving the fluctuations of diffusion coefficient,component enthalpy and species concentration. Under certain circumstances, thefluctuating components could be an important component of the diffusion process.However, adequate models are not available within the existing turbulence model toaccount for these effects. Thus, only the mean component is retained in the current versionof ANSYS CFX.

The implemented conservation of energy equation for multicomponent fluids involves onlymean scalar components and is expressed as:

(Eqn. 132)

&i i

1 %iX Y⁄

µ cv

cp +

66x j-------- $ihi

6Y i6x j--------

i

N C

V

66x j-------- $i $i

WW+( ) hi hiWW+( )6 Y i Y i

WW+( )6x j

--------------------------i

N C

V

66t----- %H( ) 6P

6t------– 66x j-------- %U jH( )+

66x j-------- +6T

6x j-------- $ihi

6Y i6x j--------

i

N C

VµtPrt------ 6h

6x j--------+ +

2 38 94 5

SE+=

Basic Solver Capability Theory: Additional Variables


Multicomponent Energy Diffusion

The energy equation can be simplified in the special case that all species diffusivities are thesame 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 beassembled, rather than one for each component plus one for heat conduction. This cansignificantly reduce numerical cost, in particular when the fluid consists of a large numberof 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

$i $ +cp-----= =

Lei + cp$i( )⁄ 1= = Lei 1=

Sci µ $i⁄ 1= = Lei 1=

t66 %H( ) P6

t6------– x j66 %U jH( )+ x j6

6 +cp-----

µtPrt-------+2 3

4 5 h6x j6------- SE+=

%0( )6t6-------------- %U0( )C•+ %D#C0( )C• S0+=

%

#

0 # %⁄=

Basic Solver Capability Theory: Additional Variables


• 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 S/%'/+"EZ" [Q("+, (G. 69 HEIJKLML N#OPLQ+R"% ST"Q%U $/H("I) 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-reactingscalar) 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.


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.

S0

D#

%0( )6t6-------------- %U0( )C•+ %D#

µtSct-------+2 3

4 5C02 34 5C• S0+=

Sct

µt

%0( )6t6-------------- %D#C0( )C• S0+=

%

#

0 # %⁄=

S0

D#

%D#C0( )C• S0+ 0=

%

#

0 # %⁄=

Basic Solver Capability Theory: Rotational Forces



unit time

• is the kinematic diffusivity for the scalar.

Algebraic Equation

Additional variables may be set up as either algebraic equations or transport equations. Foralgebraic additional variables, you must provide an expression for its value throughout thedomain. An algebraic additional variable definition can be as simple as a copy of an existingvariable.

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 forceand 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.

S0

D#

-

SM ,rot SCor Scfg+=

SCor 2%- UA–=

Scfg %- - rA( )A–=

r U

I

I hstat12--U

2 12---

2R2–+=

-

Basic Solver Capability Theory: Sources


Alternate Rotation Model

By default, the advection term in the momentum equation models the relative framevelocity:

(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 ofthe momentum equation becomes:

(Eqn. 144)

Modeling details and recommendations on when this should be used are available. Fordetails, 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 apoint within a domain. A point source is actually implemented as a volumetric source withina 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, directionallosses in porous regions, or other processes. These situations are described further in thefollowing sections. More information on using the momentum source models in ANSYS CFXis available. For details, see General Momentum Source (p. 28 in "ANSYS CFX-SolverModeling Guide").

%U6t6----------- %U UD( )C•+

p(– µ CU CU( )T+( )+( ) 2%- UA %- - rA( )A––C•=

Uabs U

%U6t6----------- %U UabsD( )C•+

p(– µ CU CU( )T+( )+( ) %- UA %- - rA( )A––C•=



Isotropic LossModel

The momentum loss through an isotropic porous region can be formulated usingpermeability 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 orperforated plate) may be modeled using the directional loss model. With this model, thestreamwise 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

%2--- U U x–=

SM y,µ

Kperm--------------U y– K loss

%2--- U U y–=

SM z,µ

Kperm--------------U z– K loss

%2--- U U z–=

K perm K loss

CR1 CR2

CR1µ

Kperm--------------=

CR2 K loss%2---=

x' y' z', , x'y' z',

SM x',µ

KpermS--------------U x'– K loss

S %2--- U U x'–=

SM y',µ

KpermT--------------U y'– K loss

T %2--- U U y'–=

SM z',µ

KpermT--------------U z'– K loss

T %2--- U U z'–=

K permS K perm

T K lossS

K lossT



In addition, the transverse loss may be modeled by multiplying the streamwise coefficientsby some factor. If this option is used together with a streamwise permeability, the impliedtransverse permeability is equal to the streamwise permeability divided by this factor. Fordetails, 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 notcovered by the isotropic or directional loss models. A different source can be specified foreach 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 ispresent.

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 theenergy 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,

6Sm

6U---------

S0

6S060--------

SMS

Basic Solver Capability Theory: Boundary Conditions


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


• 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)

0

SMS0 max SMS 0,( )0MS min SMS 0,( )0+=

0MS SMS

Di

U Inlet U speci V spec j W speck+ +=



Cylindrical Velocity ComponentsIn this case the velocity boundary condition is specified in a local cylindrical coordinatesystem. Only the axial direction of the local coordinate system needs to be given and thecomponents of velocity in the r, theta and z directions are automatically transformed by theANSYS 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 fromthe cylindrical components to the Cartesian components such that the boundary conditionis the same as if Cartesian components were specified:

(Eqn. 157)

For details, see Cylindrical Velocity Components (p. 55 in "ANSYS CFX-Solver ModelingGuide").

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 pressureinstead.

The direction constraint for the Normal To Boundary option is the same as that for theNormal Speed In option. Alternatively, the direction vector can be specified explicitly interms of its three components. In both cases, the boundary mass flow is an implicit result ofthe flow simulation.

Mass Flow RateThe boundary mass flow rate is specified along with a direction component. If the flowdirection is specified as normal to the boundary, a uniform mass influx is assumed to existover the entire boundary. Also, if the flow direction is set using Cartesian or cylindricalcomponents, the component normal to the boundary condition is ignored and, again, auniform mass influx is assumed. The mass influx is calculated using:

(Eqn. 158)

where

(Eqn. 159)

U Inlet U r spec, r U\ spec, \ U z spec, z+ +=

U Inlet U speci V spec j W speck+ +=

ptot

%U mAd

S7----------=

AdS7



is the integrated boundary surface area at a given mesh resolution. The area varies withmesh 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 equatedto 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 iscalculated using:

(Eqn. 166)

and the turbulence dissipation calculated using:

(Eqn. 167)

%U

k "k "

I

I uU----=

k "

QInletk Qadvect

k Qdiffusk+=

QInlet" Qadvect

" Qdiffus"+=

k "

Qadvectk mkspec=

Qadvect" m"spec=

I uU---- 0.037= =

kInlet32--I

2U 2=

"Inlet %Cµk2

µt-----=



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 energyand 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 stresscomponents are:

(Eqn. 173)

and the shear stress components are equal to zero:

(Eqn. 174)

µt 1000Iµ=

k "

kInlet32--I

2U 2=

"Inletk

32--

lt-----=

k "

kInlet kspec=

"Inlet "spec=

k "k

uxux uyuy uzuz13--k= = =

uxuy uxuz uyuz 0= = =



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, andequated to zero.

Total TemperatureThe boundary advection and diffusion terms for specified total temperature are evaluatedin exactly the same way as specified static temperature, except that the static temperatureis 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, seeIdeal 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--------–=



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 tozero.

Inlet (supersonic)

Mass andMomentum

At a supersonic Inlet boundary, all quantities must be specified. Static pressure, Cartesiancomponents 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 witha value for the relative static pressure.

The velocity components can be Cartesian or cylindrical in the same way as for a subsonicinlet. 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)

#Inlet #spec=

#

QInlet# Qadvect

# Qdiffus#+=

#

Qadvect# m#spec=

T stat, Inlet T spec=

T tot, Inlet T spec=

pstat, Outlet pspec=



Normal SpeedThe magnitude of the outlet velocity is specified and the direction is taken to be normal tothe boundary at mesh resolution.

Cartesian Velocity ComponentsThe boundary velocity components are specified, with a non-zero resultant out of thedomain.

(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 specifiedvalue:

(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 differencebetween the local nodal value and the average outlet boundary pressure. In this way the exitboundary condition pressure profile can float, but the average value is constrained to thespecified value.

Average Static Pressure: Above or Below Specified RadiusIn this case, the average pressure is only constrained in the region above or below thespecified radius by shifting the calculated pressure profile by the difference between thespecified average and the nodal average above or below the specified radius.

Average Static Pressure: CircumferentialThe circumferential averaging option divides the exit boundary condition intocircumferential bands (oriented radially or axially depending on the geometry). Thepressure within each band is constrained to the specified average pressure value the sameway as is done for the overall averaging:

(Eqn. 190)

where the specified value is applied within a band and the nodal average pressure is alsocalculated within a band.

UOutlet U speci V spec j W speck+ +=

pspec1A---- pn Ad

S7=

pip pspec pnode pnode–( )+=

pip pspec Tk( ) pn pn Tk( )–( )+=



Mass Flow Rate: Scale Mass FlowsThe mass flux distribution across the outlet is determined by starting with the local massflow 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 outletboundary condition:

(Eqn. 192)

where the summation is over all boundary integration points. A scaling factor is computedat the end of each coefficient loop which is used to scale the local integration point massflows 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 bythe scaling factor:

(Eqn. 194)

In this way, the mass flux profile is an implicit result of the solution and at the same timegives 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 equationto get the specified mass flow rate. Generally speaking, the mass flow rate at each boundaryintegration point is dependent upon both velocity and pressure:

(Eqn. 195)

where the integration point velocity depends upon nodal velocity and integration pointpressures through the Rhie-Chow coupling. For this boundary condition, the integrationpoint pressures are given by an expression of the form:

(Eqn. 196)

mip˙ %ipAipU ip=

mtotest mip

˙allV=

Fmspec

mtotest------------=

F

mip˙ F%ipAipU ip=

mip˙ %ipAipU ip pip U node,( )=

pip F pprof 1 F–( ) pnode pnode–( ) pshift+ +=



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 loopto 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 (oroutlet boundaries using an Average Static Pressure specification) enforces the specifiedprofile as an average pressure profile (or average pressure) in circumferential bands (radialor axial), held at a particular value.

Starting with the original formula for the integration point pressure in (Eqn. 196): instead ofimposing a particular profile distribution, an average pressure profile within bands isintroduced:

(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 accompanyingvalues, except for the radiation intensity if radiation is modeled.

pprof pnode

pnode F

pshift

mspec %ipV ipAipU ip pip U node,( )=

pip F pprof Tk( ) pnode pnode– Tk( )( ) pshift+ +=

pprof Tk( ) Tk pnode Tk( )

Tk F

pprof Tk( )



Opening

An opening boundary condition allows the fluid to cross the boundary surface in eitherdirection. For example, all of the fluid might flow into the domain at the opening, or all ofthe fluid might flow out of the domain, or a mixture of the two might occur. An openingboundary condition might be used where it is known that the fluid flows in both directionsacross 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 foroutflow.

The direction component, normal to boundary condition or direction components, mustalso be specified. The magnitude of the velocity at the opening boundary is then part of thesolution.

You can optionally specify a loss coefficient, :

(Eqn. 201)

with a velocity component specification at an opening boundary. The pressure drop iscalculated 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=

Bploss12-- f%U n

2=

U n

pspec12-- f%U n

2– pstat=



and for outflows, the constraint becomes:

(Eqn. 204)

The coefficient can be used to model the pressure drop across a screen or other planarresistance.

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 thatspecified 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 thedomain is calculated by the ANSYS CFX-Solver and is the additional variable value takenfrom 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 thevelocity becomes:

(Eqn. 207)

Free SlipIn this case, the velocity component parallel to the wall has a finite value (which iscomputed), but the velocity normal to the wall, and the wall shear stress, are both set tozero:

(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 arethree different options for the wall velocity:

pspec12-- f%U n

2+ pstat=

T stat,Opening T spec=

#Opening #spec=

UWall 0=

Un,Wall 0=

.w 0=



• 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 thedomain 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 framesby explicitly setting the angular velocity equal to minus one times the domain angularvelocity.

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 U\ spec, \ U z,specz+ +=

UWall -R–=

R -

UWall -R=

qw 0=

T stat,w T spec=

Basic Solver Capability Theory: Automatic Time Scale Calculation


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 "ANSYSCFX-Solver Modeling Guide").

AdditionalVariables

The options for specifying additional variable quantities at wall boundaries are analogousto those for heat transfer.

Symmetry Plane

The symmetry plane boundary condition imposes constraints which ‘mirror’ the flow oneither 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 timestepused 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=

n660 0=



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 Optionparameter, 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=

UBPpbc,max pbc,min–

%node--------------------------------------=

U bc U node

pbc,max pbc,min

%node

Mmax U bc U node UBP, ,( )

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

c 6%6p------2 3

4 51–

=

c



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 soliddomain.

A volume averaged diffusion is calculated as:

(Eqn. 230)

g g=

g g ' Tmax Tmin–( )=

' Tmax Tmin

Bt min BtU BtBP Bt g Btrot Btc, , , ,( )=

BtU 0.3Lscale

max U bc U node,( )-----------------------------------------= BtBP 0.3LscaleUBP------------=

Bt gLscale

g------------= Btrot0.1--------=

M 0.3>

BtcLbc

max U bc U node UBP c, , ,( )------------------------------------------------------------=

Lscale

& +%cp--------=

Basic Solver Capability Theory: Mesh Adaption


where each quantity is volume averaged over the solid. The solid time scale is finallycalculated 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 meshis selectively refined in areas which depend on the adaption criteria specified. This meansthat as the solution is calculated, the mesh can automatically be refined in locations wheresolution variables are changing most rapidly, in order to resolve the features of the flow inthese regions.

More information on the mesh adaption process and controls is available. For details, seeMesh Adaption (p. 205 in "ANSYS CFX-Pre User's Guide"). Within the mesh adaption stepitself, 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 newmesh.

This section describes the details of exactly which elements are refined and how therefinement 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.

Bt fLscale

2

&------------=

f

Ai

i li

Ai0 jiB

N0 j0 jB--------------------

jV=

0 j jth B0 j

0 j

B0ij 0 j

N0 jj Ai



• 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 evaluatedat each node, and the calculation of the adaption criteria follows as if this were anothersolution variable.

The edges which have the largest adaption criteria are marked for refinement, providingthat this would not result in mesh edges which were shorter than the Minimum EdgeLength, if this was specified.

Mesh Refinement Implementation in ANSYS CFX

There are two general methods for performing mesh adaption. Incremental adaptiontakes an existing mesh and modifies it to meet the adaption criteria. The alternative isre-meshing, in which the whole geometry is re-meshed at every adaption step accordingto the adaption criteria. In ANSYS CFX, incremental adaption is used, since this is muchfaster, however, this imposes the limitation that the resulting mesh quality is limited by thequality of the initial mesh.

The particular implementation of incremental adaption that is adopted in ANSYS CFX isknown as hierarchical refinement or h-refinement. Each adaption step consists of astructured refinement of an existing mesh. A sequence of refinements form a set ofhierarchical levels.

In each mesh adaption step, each mesh edge which is marked for adaption (see the previoussection) has an extra node placed half-way along it. The mesh elements which share thisedge 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 0 jiB

N0 j0 jB--------------------

jV=

0 jB jth 0 j

0 jiB

0 j N0 j

Ai



• 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 pyramidelements near wall boundary conditions), the mesh adaption avoids refining theseelements in the direction perpendicular to the wall. Only edges on the interface betweenthe inflated elements and the rest of the tetrahedral mesh are allowed to be marked foradaption. When the refinement of these edges takes place, the refinement propagatesthrough 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 theoriginal geometry. This means that nodes which are added to the surface of the problem areadded onto the surfaces of the original geometry rather than to the edges of the existingmesh. 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



When using Adapt to Geometry, the adaption process can help to resolve fine features inthe geometry. When this is done, nodes near the surface may be moved slightly in order tomake sure that the quality of the resulting elements is not degraded (not shown in the figureabove).

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 solutionby refining the mesh in front of or behind the discontinuity, then your solution was notsufficiently converged before mesh adaption took place. As a solution containing adiscontinuity develops, the location of such a discontinuity may move. If the solution isnot sufficiently converged before mesh adaption takes place, then the mesh will be

Basic Solver Capability Theory: Flow in Porous Media


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 tabto correct this problem.

Flow in Porous Media

Flow in porous media in ANSYS CFX can be calculated using either a model for momentumloss or a full porous model. The momentum loss model is available in fluid domains, whilethe 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 ofDarcy's law commonly used for flows in porous regions. It can be used to model flows wherethe geometry is too complex to resolve with a grid. The model retains both advection anddiffusion terms and can therefore be used for flows in rod or tube bundles where sucheffects are important.

In deriving the continuum equations, it is assumed that ‘infinitesimal’ control volumes andsurfaces are large relative to the interstitial spacing of the porous medium, though smallrelative to the scales that you wish to resolve. Thus, given control cells and control surfacesare 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)

1 V 'V

V ' 1V=

A'A

A' K A:=

K K ij( )=

K Ai: K ijA j= K

t66 1%#( ) %K U: #( )C• $K #C:( )C•–+ 1S=



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 forpossible anisotropies in the resistance.

In the limit of large resistance, a large adverse pressure gradient must be set up to balancethe resistance. In that limit, the two terms on the r.h.s. of (Eqn. 239) are both large and ofopposite 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, withpermeability 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

t66 1% %K U:( )C•+ 0=

t66 1%U( ) % K U:( ) UD( )C• µeK UC UC( )T+( ):( )C•–+

1R U:– 1 pC–=

U µe

R Rij( )=

U R 1– pC:–=

U

Q K U:=

t66 1%H( ) %K U: H( )C• $eK HC:( )C•–+ 1SH=

$e SH



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 usessuperficial velocity. If so, the coefficients are represented by:

(Eqn. 246)

where is the porosity.

6 %U i( )6t-----------------

6 %U jU i( )6x j

------------------------+ 6p6xi-------– %gi

6. ji6x j--------- Si

M+ + +=

SiM

SiM CR1U i– CR2 U U i– Si

spec+=

CR1

CR2

Sispec

U U

6p6xi-------– µ

K perm--------------U i K loss

%2--- U U i+=

µ

K perm

K loss

CR1 µK---- CR2, K loss

%2---= =

CR1 µ1K------- CR2,

K loss%

21 2--------------= =

1





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 complexprocess, mainly because it is three dimensional, unsteady and consists of many scales. It canhave a significant effect on the characteristics of the flow. Turbulence occurs when theinertia forces in the fluid become significant compared to viscous forces, and ischaracterized by a high Reynolds Number.

In principle, the Navier-Stokes equations describe both laminar and turbulent flows withoutthe need for additional information. However, turbulent flows at realistic Reynolds numbersspan a large range of turbulent length and time scales, and would generally involve lengthscales much smaller than the smallest finite volume mesh, which can be practically used ina numerical analysis. The Direct Numerical Simulation (DNS) of these flows would requirecomputing power which is many orders of magnitude higher than available in theforeseeable future.

Turbulence and Wall Function Theory: Turbulence Models


To enable the effects of turbulence to be predicted, a large amount of CFD research hasconcentrated on methods which make use of turbulence models. Turbulence models havebeen specifically developed to account for the effects of turbulence without recourse to aprohibitively fine mesh and direct numerical simulation. Most turbulence models arestatistical turbulence model, as described below. The two exceptions to this in ANSYS CFXare 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 additionaltime-varying, fluctuating component. For example, a velocity component may be dividedinto an average component, and a time varying component.

In general, turbulence models seek to modify the original unsteady Navier-Stokes equationsby the introduction of averaged and fluctuating quantities to produce the ReynoldsAveraged Navier-Stokes (RANS) equations. These equations represent the mean flowquantities only, while modeling turbulence effects without a need for the resolution of theturbulent fluctuations. All scales of the turbulence field are being modeled. Turbulencemodels based on the RANS equations are known as Statistical Turbulence Models due to thestatistical averaging procedure employed to obtain the equations.

Simulation of the RANS equations greatly reduces the computational effort compared to aDirect Numerical Simulation and is generally adopted for practical engineering calculations.However, the averaging procedure introduces additional unknown terms containingproducts of the fluctuating quantities, which act like additional stresses in the fluid. Theseterms, called ‘turbulent’ or ‘Reynolds’ stresses, are difficult to determine directly and sobecome further unknowns.

The Reynolds (turbulent) stresses need to be modeled by additional equations of knownquantities in order to achieve “closure.” Closure implies that there is a sufficient number ofequations for all the unknowns, including the Reynolds-Stress tensor resulting from theaveraging procedure. The equations used to close the system define the type of turbulencemodel.

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


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 averagingis actually weighted by density (Favre-averaging), but for simplicity, the followingpresentation assumes that density fluctuations are negligible.

For transient flows, the equations are ensemble-averaged. This allows the averagedequations to be solved for transient simulations as well. The resulting equations aresometimes called URANS (Unsteady Reynolds Averaged Navier-Stokes equations).

Substituting the averaged quantities into the original transport equations results in theReynolds-averaged equations given below. For details, see Transport Equations (p. 23). Inthe following equations, the bar is dropped for averaged quantities, except for products offluctuating quantities.

(Eqn. 3)

(Eqn. 4)

where is the molecular stress tensor.

The continuity equation has not been altered but the momentum and scalar transportequations 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 thatconvective transport due to turbulent velocity fluctuations will act to enhance mixing overand above that caused by thermal fluctuations at the molecular level. At high Reynoldsnumbers, turbulent velocity fluctuations occur over a length scale much larger than themean free path of thermal fluctuations, so that the turbulent fluxes are much larger than themolecular fluxes.

The Reynolds-averaged energy equation is:

(Eqn. 5)

U 1Bt------ U td

t

t Bt+

7=

Bt

%6t6------ %U( )C•+ 0=

%U6t6------------ %U UD{ }C•+ . %– u uD{ } SM+C•=

.

%u uD %u0

%htot6t6--------------

p6t6------– %Uhtot( )C•+ +C. %uh–( ) C U .•( ) SE+•+C•=

Turbulence and Wall Function Theory: Eddy Viscosity Turbulence Models


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 ANSYSCFX-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 thecomputation of the Reynolds stresses and Reynolds fluxes. ANSYS CFX models can bebroadly 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 continuouslyforming and dissipating, and in which the Reynolds stresses are assumed to be proportionalto mean velocity gradients. This defines an ‘eddy viscosity model.’

The eddy viscosity hypothesis assumes that the Reynolds stresses can be related to themean 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 laminarNewtonian flow:

(Eqn. 9)

where is the Eddy Viscosity or Turbulent Viscosity. This has to be modeled.

%uh

C U .•( )•

htot h 12--U

2 k+ +=

k 12--u

2=

%06t6--------- %U0( )C•+ $C0 %– u0( ) S0+C•=

%u uD– µt CU CU( )T+( ) 23--( %k µtC U:+( )–=

µt



Analogous to the eddy viscosity hypothesis is the eddy diffusivity hypothesis, which statesthat 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 transportequations 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 TheoryGuide").) is neglected in the solver and thus not included here.

%u0– $tC0=

$t

$tµtPrt-------=

Prt

µt k " k -

%U6t6------------ %U UD( )C•+ B Cp'– µeff CU CU( )T+( )( )C•+=

B µeff

µeff µ µt+=

p'

p' p 23--%k 2

3--µtC U:+ +=

p' p= 23--%k

u 23--(C U•2 34 5–



The Reynolds averaged energy equation becomes:

(Eqn. 15)

Note that although the transformation of the molecular diffusion term may be inexact ifenthalpy depends on variables other than temperature, the turbulent diffusion term iscorrect, subject to the eddy diffusivity hypothesis. Moreover, as turbulent diffusion is usuallymuch larger than molecular diffusion, small errors in the latter can be ignored.

Similarly, the Reynolds averaged transport equation for Additional Variables (non-reactingscalars) becomes:

(Eqn. 16)

Eddy viscosity models are distinguished by the manner in which they prescribe the eddyviscosity 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 transportequations are solved, these models are termed ‘zero equation.’

The zero equation model in ANSYS CFX uses an algebraic equation to calculate the viscouscontribution from turbulent eddies. A constant turbulent eddy viscosity is calculated for theentire 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.

%htot( )6t6------------------- p6

t6------ %Uhtot( )C•+– + .µtPrt------- hC+C2 3

4 5 C U .•( ) SE+•+C•=

%06t6--------- %U0( )C•+ $0

µt,0------+2 3

4 5C0C• S0+=

µt

U t

lt

µt % f µU tlt=

f µ

lt V D

13--

2 38 94 5

7⁄=

V D



Two Equation Turbulence Models

Two-equation turbulence models are very widely used, as they offer a good compromisebetween numerical effort and computational accuracy. Two-equation models are muchmore sophisticated than the zero equation models. Both the velocity and length scale aresolved 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 turbulentviscosity 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 turbulentkinetic energy, which is provided from the solution of its transport equation. The turbulentlength scale is estimated from two properties of the turbulence field, usually the turbulentkinetic energy and its dissipation rate. The dissipation rate of the turbulent kinetic energy isprovided 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 " k -

"k

k "

%6t6------ %U( )C•+ 0=

%U6t6------------ %U UD( ) µeff UC( )C•–C•+ p'C– µeff UC( )TC• B+ +=

B µeff

p'

k "

µeff µ µt+=



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 CompressibleProduction (accessible on the Advanced Control part of the Turbulence section in ANSYSCFX-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 "

µt Cµ%k2

"-----=

Cµ

k "

%k( )6t6-------------- %Uk( )C•+ µ

µt,k-----+2 3

4 5 kCC• Pk %"–+=

%"( )6t6-------------- %U"( )C•+ µ

µt,"-----+2 3

4 5 "CC• "k--- C"1Pk C"2%"–( )+=

C"1 C"2 ,k ,"

Pk

Pk µt U UC UC T+( )• 23-- U 3µt UC• %k+( )C•–C Pkb+=

UC•

UC•

3µt

k "

µt

Pkb

Pkbµt%,%---------– g %C•=



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%,%---------%'g TC•=

k"

Pkb

P"b C3 max 0 Pkb,( ):=

P"b

&

P"b C3 max 0 Pkb,( ) &sin::=

,% ,%

-Pkb P"b " ''-k=

k--

k " C"1

C"1RNG

%"( )6t6-------------- %U"( )C•+ µ

µt,"RNG---------------+2 3

4 5 "CC• "k--- C"1RNGPk C"2RNG%"–( )+=

C"1RNG 1.42 f T–=



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 T

T 1 T4.38----------–2 3

4 5

1 'RNGT3+( )

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

TPk

%CµRNG"-----------------------=

=

k -

k "

k " y+ 0.2<

k - y+ 2<

y+ 2<k -

k -

µt % k-----=

k -k

-

k

%k( )6t6-------------- %Uk( )C•+ µ

µt,k-----+2 3

4 5CkC• Pk 'W%k-–+=

-

%-( )6t6--------------- %U-( )C•+ µ

µt,-------+2 3

4 5C-C• &-k----Pk '%-2–+=



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, twoproduction 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 DistanceFormulation (p. 117). At the boundary layer edge and outside the boundary layer, the

standard - model is therefore recovered.

% UPk

k "

'W 0.09=

& 5 9⁄=

' 0.075=

,k 2=

,- 2=

.

. µt2s %23--(k–=

Pkb k

-

-

k - k "k "

k -F1 k "

1 F1– F1

k "



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 correspondingcoefficients of the underlying models:

(Eqn. 48)

%k( )6t6-------------- %Uk( )C•+ µ

µt,k1--------+2 3

4 5CkC• Pk 'W%k-–+=

%-( )6t6--------------- %U-( )C•+ µ

µt,-1---------+2 3

4 5C-C• &1-k----Pk '1%-

2–+=

k-"

%k( )6t6-------------- %Uk( )C•+ µ

µt,k2--------+2 3

4 5CkC• Pk 'W%k-–+=

%-( )6t6--------------- %U-( )C•+

µµt,-2---------+2 3

4 5C-C• 2% 1,-2 ---------------- k -CC &2

-k----Pk '2%-

2–+ +2 34 5=

F1 k "

1 F1– k -

%k( )6t6-------------- %Uk( )C•+ µ

µt,k3--------+2 3

4 5CkC• Pk 'W%k-–+=

%-( )6t6--------------- %U-( )C•+

µµt,-3---------+2 3

4 5C-C• 1 F1–( )2% 1,-2 ---------------- k -CC &3

-k----Pk '3%-

2–+ +

#3 F1#1 1 F1–( )#2+=



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 adversepressure 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. Thereasons for this deficiency are given in detail in Menter [9]. The main reason is that bothmodels do not account for the transport of the turbulent shear stress. This results in anoverprediction of the eddy-viscosity. The proper transport behavior can be obtained by alimiter 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.

'W 0.09=

&1 5 9⁄=

'1 0.075=

,k1 2=

,-1 2=

&2 0.44=

'2 0.0828=

,k2 1=

,-2 1 0.856⁄=

k -

k "

vta1k

max a1- SF2,( )--------------------------------------=

vt µt %⁄=

F2 F1

S



Blending FunctionsThe blending functions are critical to the success of the method. Their formulation is basedon 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 theANSYS 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 productionterm 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 consistentlyavoided the stagnation point build-up in aerodynamic simulations.

F1 ar g14( )tanh=

ar g1 min max k'W-y------------ 500E

y2-------------,

2 38 94 5 4% k

CDkw,-2 y2------------------------------,2 38 94 5

=

y E

CDk- max 2% 1,-2-------------- k - 1.0 10 10–A,CC2 3

4 5=

F2 ar g22( )tanh=

ar g2 max 2 k'W-y------------ 500v

y2------------,

2 38 94 5

=

k " k -

Pk

Pk min Pk Clim%",( )=

Clim -



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 " 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 theproduction 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 i6xj6---------

U j6xi6--------+2 3

4 5=

S Sij

S

^

Pk µtS^=

^ 2îjîj= îj, 12--

U i6xj6---------

U j6xi6--------–2 3

4 5=

^ îj

S ^

-

k-" k-"( )1E

6%vt6t-----------

6%U jvt6x j

------------------+ c1%vt S c2%vt

LvK--------2 3

4 5 2µ

%vt,--------+2 3

4 5 6v6x j--------+–=

v vt ,



The model contains a destruction term, which accounts for the structure of turbulence andis 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 goesto 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

6S6x j-------- 6S6x j

-------------------------=

µt %vt=

Ek "–vt

LvK--------2 3

4 5 2=

EBB6vt6x j--------

6vt6x j--------=

E1e c3EBBEk "–c3EBB--------------2 3

4 5tanh=

6%vt6t-----------

6%U jvt6x j

------------------+ c1D1%vtS c2%E1e µ%vt,--------+2 3

4 5 6vt6x j--------+–=

*

,

Turbulence and Wall Function Theory: Reynolds Stress Turbulence Models


Low ReynoldsNumberFormulation

Low Reynolds formulation of the model is obtained by including damping functions. Nearwall 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 stresstensor and the dissipation rate. These models do not use the eddy viscosity hypothesis, butsolve an equation for the transport of Reynolds stresses in the fluid. The Reynolds stressmodel 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 equationsindividually for each Reynolds stress component.

The exact production term and the inherent modeling of stress anisotropies theoreticallymake Reynolds Stress models more suited to complex flows; however, practice shows thatthey 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+*v-------------

2 38 94 5 2

–exp–=

µt %D2vt=

k-"( )1E

y+ 1<

%U6t6------------ %U UD( ) µ UC( )C•–C•+ p''C– %u uD( )C•– B+=

p''

%u uD

p'' p UC• 23--µ )–2 34 5+=



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 uD

%u uD

%u uD6t6--------------------- %u uD UD( ) %Ck

"--u uD u uDC( )T

2 34 5C•–C•+

P G 0 23--(%"–+ +2 3

4 5=

0

k-" k--

"

%u uD6t6--------------------- %U u uDD( )C•+

P 0 µ 23--cs%

k2

"-----+2 34 5 u uDC2 34 5C• 2

3--(%"–+ +2 34 5=

t66 %uiu j( ) xk6

6 U k%uiu j( )+ Pij 0ij xk66 µ 2

3--cs%k2

"-----+2 34 5 uiu j6

xk6-------------23--(ij%"–+ +=

0ij

P % u uD UC( )T UC( )u uD+( )–=

"

%"( )6t6-------------- xk6

6 %U k"( )+ "k-- c"1P c"2%"–( ) xk6

6 µµt,"-----+2 3

4 5xk6

6"+=



In these equations, the anisotropic diffusion coefficients of the original models are replacedby an isotropic formulation, which increases the robustness of the Reynolds stress model.

The Reynolds Stress model is also available with anisotropic diffusion coefficients. In thiscase, the ANSYS CFX-Solver solves the following equations for the transport of the Reynoldsstresses:

(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)

t66 %uiu j( ) xk6

6 U k%uiu j( )+

Pij 0ij xk66 µ(kl cs%

k"--ukul+2 3

4 5 uiu j6xl6-------------

23--(ij%"–+ +

2 38 94 5

=

0ij

"

%"( )6t6-------------- xk6

6 %U k"( )+ "k-- c"1P c"2%"–( ) xk6

6 µ(kl c"%k"--ukul+2 3

4 5xl6

6"+=

0ij

0ij 0ij1 0ij2+=

0ij1 %" Cs1a Cs2 aa 13--a a(•–2 3

4 5+2 34 5–=

0ij2 Cr1Pa– Cr2%kS Cr3%kS a a•–+=

Cr4%k aST SaT 23--a–+ S• (2 3

4 5 Cr5%k aWT WaT+( )+ +



and:

(Eqn. 86)

In this formulation, a is the anisotropy tensor, S is the strain rate and W is the vorticity. Thisgeneral form can be used to model linear and quadratic correlations by using appropriatevalues for the constants. The model constants are listed below.

AlternativeReynolds StressConstants

There are three varieties of the standard Reynolds stress models available. These are knownas 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” standsfor Isotropization of Production, and “QI” stands for Quasi-Isotropic. In these models, thepressure-strain correlation is linear.

The SSG model was developed by Speziale, Sarkar and Gatski [5]. This model uses aquadratic 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 Domainsform in ANSYS CFX-Pre. The following options correspond to the types of models listedabove:

• Reynolds Stress Model - LRR-IP

• SSG Reynolds Stress Model - SSG

• QI Reynolds Stress Model - LRR-IQ

a u uDk--------------- 2

3--(–=

S 12-- UC UC( )T+( )=

W 12-- UC UC( )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 "1



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 alow-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)

-

k--

- -- -

6 %.ij( )6t-----------------

6 U k%.ij( )6xk

------------------------+ %Pij– 23--'W%-k(ij %_ij– 6

6xk-------- µ

µt,*------+2 3

4 5 6.ij6xk--------2 3

4 5+ +=

-

6 %-( )6t---------------

6 U k%-( )6xk

-----------------------+ &%-k----Pk '%-2– 66xk-------- µ

µt,-----+2 3

4 5 6-6xk--------2 3

4 5+=

,` 2=, 2=' 0.075=

& ''W-----

*2

, 'W( )0.5-------------------– 59--= =

& ' -,* ,

- "-

%-( )6t6--------------- 6 U k%-( )+

&3-k----Pk '3%-

2– 66xk--------+ µ

µt,-3---------+2 3

4 5 6-6xk-------- 1 F1–( )2% 1

,2----------- 6k

6xk-------- 6-6xk

--------+2 34 5=



• 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)

-

,`1 2=,1 2=

'1 0.075=

&1''W-----

*2

, 'W( )0.5-------------------– 0.553= =

' k--*

"

,`2 1.0=,2 0.856=

'2 0.0828=

&2''W-----

*2

, 'W( )0.5-------------------– 0.44= =

-

03 F 01: 1 F–( )02+=

F arg4( )tanh=

arg min max k'W-y------------ , 500/

y2-------------

2 38 94 5

, 4% kCDk-,k "– y2-----------------------------------

; <= >? @

=

CDk- max 2% 1,k "– ------------------- 6k

6x j-------- 6-6x j

-------- , 10 10–2 34 5=

_ij

'WC1- .ij23--k(ij+2 3

4 5 & Pij23--P(ij–2 3

4 5– ' Dij23--P(ij–2 3

4 5– 1k Sij13--Skk(ij–2 3

4 5–2 34 5=



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 wallresolution.

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 beenimplemented into ANSYS CFX for use with any of the available Reynolds stress transportmodels. These terms are described in a book by Wilcox [30], and in more detail in a paper byLaunder, Tselepidakis and Younis [31].

0.09

C1 1.8C2 0.52

Pij .ik6U j6xk--------- . jk

6U i6xk--------- ;+= P 1

2--Pkk=

Dij .ik6U k6x j---------- . jk

6U k6xi----------+=

k--

µT % k-----=

'W& 8 C2+( ) 11⁄

' 8C2 2–( ) 11⁄

1 60C2 4–( ) 55⁄

-k-- k--

k-"k--

Turbulence and Wall Function Theory: ANSYS CFX Transition Model Formulation


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 areequal.

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 therotating 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 discussedbriefly 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 Reynoldsnumber equations are solved.

The models can be used with the BSL, SST and SAS-SST (beta feature) turbulence models. Inthe following the focus is on the two equation model. The empirical correlation used by thismodel has been formulated to cover standard bypass transition as well as flows in lowfree-stream turbulence environments (Langtry and Menter correlation). This model is the

^.ij

Gij % ^k . jm"ikm .im" jkm+( ): :=

"ijk

Sij 0.56U i6x j---------

6U j6xi---------+2 3

4 5=

îj

îj 0.56U i6x j---------

6U j6xi---------–2 3

4 5 "ijk ^k:+=

-pPij



recommend transition model for general-purpose applications and has been validatedtogether with the SST turbulence model [[101], [102], [103]]. It should be noted that a fewchanges have been made to this model compared to CFX-5.7 [[101],[102]) in order toimprove 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 historyeffects on the transition onset location.

It is recommended to use the new formulation [[103]], although the CFX-5.7 formulation canbe recovered by specifying the optional parameter “Transition Model Revision = 0” in theCCL 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 asfollows:

(Eqn. 105)

1

%1( )6t6--------------

%U j1( )6x j6--------------------- P11 E11– P12 E12– x j6

6 µµt,1------+2 3

4 5 16x j6-------+ +=+

P11 2Flength%S 1Fonset[ ]c1 3 E11; P111= =

S Flength

P12 2c11( )%^1Fturb E12; c12P121= =



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)

^

ReE%y2S

µ------------ RT;%kµ--------= =

Fonset1ReE

2.193 Re\c:----------------------------=

Fonset2 min max Fonset1 Fonset14,( ) 2.0,( )=

Fonset3 max 1RT2.5-------2 34 5

30,–2 3

4 5=

Fonset max Fonset2 Fonset3– 0,( )=

Fturb eRT

4------2 34 5

4–

=

Re\c

Re\t

Flength Re\c Re\t

cy1 0.03=

cy2 50=

cy3 0.5=

,y 1.0=

1 sep min 2 maxReE

3.235Re\c------------------------2 34 5 1 0,– Freattach 2,:2 3

4 5 F\t=

Freattach eRT

20------2 34 5

4–

=

1 eff max 1 1 sep,( )=



The model constants in (Eqn. 110) have been adjusted from those of Menter et al. [[101]] inorder 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 turbulenceintensity.

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

ReE Re\c

11

Re\t

%Re\t( )6t6----------------------

%U jRe\t( )6x j6-----------------------------+ P\t x j6

6 ,\t µ µt+( )Re\t6

x j6-------------2 38 94 5

+=

P\t c\t%t--- Re\t Re\t–( ) 1.0 F\t–( ) t; 500µ

%U 2------------= =

F\t min max Fwake ey(---2 34 5 4–

1.0 1 1 50⁄–1.0 1 50⁄–-------------------------2 34 5 2

–,:2 38 94 5

1.0,2 38 94 5

=

\BLRe\tµ%U-------------- (BL; 15

2-----\BL (; 50^yU------------- (BL:= = =

Re-%-y2

µ------------- Fwake; e

Re-1 5A10--------------2 34 5 2–

= =

Re\t

c\t 0.03 ,\t; 2.0= =

Re\t Re\t

Re\t

Flength



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) forthe mean flow, turbulence and transition equations.

Re\c

Re\t Flength

Re\t f Tu +,( ) Flength; f Re\t( ) Re\c, f Re\t( )= = =

Tu+\

+\\2

v-----2 34 5 Ud

sd-------=

Ud sd⁄

t66 %k( ) x j6

6 %u jk( )+ Pk Dk– x j66 µ ,kµt+( ) k6

x j6-------2 34 5+=

Pk 1 eff Pk Dk; min max 1 eff 0.1,( ) 1.0,( )Dk= =

Ry%y k

µ------------- F3; eRy

120--------2 34 5

8–

F1; max Florig F3,( )= = =

Pk Dk

F1orig

-

y+ y+

y+

Turbulence and Wall Function Theory: Large Eddy Simulation Theory


Large Eddy Simulation Theory

This section outlines the theoretical details of the LES model in ANSYS CFX. Additionalinformation on setting up an LES simulation is available, as well as modeling advice. Fordetails, see The Large Eddy Simulation Model (LES) (p. 111 in "ANSYS CFX-Solver ModelingGuide").

The non filtered Navier-Stokes equations are:

(Eqn. 122)

Large Eddy Simulation (LES) is about filtering of the equations of movement anddecomposition 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 filteredNavier-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 nolonger true.

6 %U i( )6t-----------------

6 %U iU j( )6x j

------------------------+ 6p6xi-------– µ

62U i6x j6x j----------------+=

f f f '+=

f

f xi t,( ) G xi xi'–( ) f xi' t,( ) xi'dVol7=

G xi xi'–( )

6 %U i( )6t-----------------

6 %U iU j( )6x j

------------------------+ 6p6xi-------– µ

62U i6x j6x 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)



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 averagingassumptions 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.

.ij

.ij uiu j U iU j–=

6 %U i( )6t-----------------

6 %.ij %U iU j+( )6x j

----------------------------------------+ 6p6xi-------– µ

62U i6x j6x j----------------+=

6 %U i( )6t-----------------

6 %U iU j( )6x j

------------------------+ 6p6xi-------– µ

62U i6x j6x j----------------

6 %.ij( )6x j

-----------------–+=

.ij 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

.ij13--.kk– 2 vSGS Sij: :– v– SGS

6U i6x j---------

6U j6xi---------+

2 38 94 5

:= =

vSGS l qSGSa

B Vol( )1 3⁄=



In the Smagorinsky model, based on an analogy to the Prandtl mixing length model, thevelocity 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 B S= where S 2SijSij( )1 2⁄=

vSGS CSB( )2 S=

E k( ) Ck"2 3⁄ k 5 3⁄–=

CS1G---

23Ck---------2 3

4 5 3 4⁄0.18= =

f µ

µT % min lmix f µCSB,( )2 2SijSij:=

lmix * ywall:=

*

f µ

f µ 1 y A⁄–( )exp–=

Turbulence and Wall Function Theory: Detached Eddy Simulation Theory


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 modelis 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 onsetting up a DES simulation and modeling advice are available. For details, see TheDetached 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 isprohibitively 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 structuregenerating them (airfoil flaps, buildings, blunt trailing edges on turbine blades). DES is anattempt to combine elements of RANS and LES formulations in order to arrive at a hybridformulation, where RANS is used inside attached and mildly separated boundary layers.Additionally, LES is applied in massively separated regions. While this approach offers manyadvantages, it is clearly not without problems, as the model has to identify the differentregions automatically. DES models require a detailed understanding of the method and thegrid generation requirements and should not be used as “black-box.” You are advised toread the CFX technical report [55] on the subject, which explains all details on gridgeneration requirements, zonal formulation and boundary conditions.

The ANSYS CFX version of the DES model is based on the SST formulation. The advantage ofthis combination is that the accurate prediction of turbulent boundary layers up toseparation and in mildly separated regions carries over from the SST model. In addition, theSST model supports the formulation of a zonal DES formulation [56], which is less sensitiveto 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 Eu



SST-DES Formulation Strelets et al.

The idea behind the DES model of Strelets [58] is to switch from the SST-RANS model to anLES 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 thedissipation 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 thatthe model should return the RANS formulation in attached boundary layers. The maximumedge 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 inthe 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 differencescheme 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 thestandard SST-DES model) is that there is no mechanism for preventing the limiter frombecoming 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 CFXoffers a zonal formulation of the DES formulation, based on the blending functions of theSST model [56]. The blending functions are functions of the wall distance. For details, seeWall Distance Formulation (p. 117).

(Eqn. 145)

B

" '`k- k32-- Lt⁄ k

32-- CDESB( )⁄ for CDESB Lt<( )b= =

B max Bi( ) ; Lt k( ) '⁄ `-= =

" '`k- '`k- FDES with FDES maxLt

CDESB---------------- 1( , )=:b=

k "–

Bs c(<

FDES CFX– maxLt

CDESB---------------- 1 FSST–( ) 1( , ) with FSST 0 F1 F2, ,==



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 alsobe 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 schemein 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 betweenthe 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=

c

C %Uc: 1^CV----------- mipcip

ipVd

^CV ip

mip cip

ip cip

cip U, cip C, ,

cip , cip U, 1 ,–( ) cip C,:+:=

,

, ,max ACH1( )tanh:=

A CH2 maxCDESBmax

Lturb g------------------------ 0.5– 0,2 34 5:=

Lturbk

cµ----------=

g max B4( )tanh 10 10–,( )=

B CH3^max ^ S,( )

max S2 ^2+2------------------ 10 10–,2 3

4 5---------------------------------------------------:=

Turbulence and Wall Function Theory: Scale-Adaptive Simulation Theory


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 twoblending functions. The default and the recommendation is to use the second SST blendingfunction 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 oscillationsdue to the central difference scheme, which can occur for medium and highCourant-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 inletboundaries. This greatly complicates the use of LES in industrial flows, as the details of thesefluctuations are generally not known. In DES simulations, it is in most cases possible tospecify the same boundary conditions as for the SST model. This is particularly true if thedefault setting of the zonal model are used ([55]).

Scale-Adaptive Simulation Theory

The Scale-Adaptive Simulation (SAS) is an improved URANS formulation, which allows theresolution of the turbulent spectrum in unstable flow conditions. The SAS concept is basedon the introduction of the von Karman length-scale into the turbulence scale equation. Theinformation provided by the von Karman length-scale allows SAS models to dynamicallyadjust to resolved structures in a URANS simulation, which results in a LES-like behavior inunsteady regions of the flowfield. At the same time, the model provides standard RANScapabilities in stable flow regions.

Bmax

,max 1= CH1, 3 CH2, 1 CH3, 2= = =

,

,SST-DES max , BF1 F1 BF2 F2:,:,( )=

F1 F2

,

,lim max , 1 min CFLmax

CFL------------------ 1,2 34 5

CFLEXP–,=

CFLmax CFLEXP



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 theunderlying 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 transportequation 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-"( )1E

k-"( )1E

k-"

LvK

LvK

LvK

k-Et

k-Et

-

k-Et

k # kL=

%k6t6---------

U j%k6x j6----------------+ Pk cµ

3 4⁄ %k2

#-----6y6------

µt,k------ k6

y6------+–=

%#6t6-----------

U j%#6x j6------------------+ )1

0k---Pk )2µtS Ue #2

k3 2⁄---------- )3%k 6y6------

µt,#------- #6

y6-------+––=

vt cµ1 4⁄ #=

Ue62U i

6x j2-----------62U i

6x j2-----------=



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 foundationand 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 isproportional to the size of the resolved eddies. Standard turbulence models, on the otherhand, always provide a length-scale proportional to the thickness of the shear layer. They donot 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= *, 0.41=

)1 0.8=

)3 0.0326= )2 3.51=

,k ,# 2 3⁄= =

# UeL

LvK * U6 y6⁄62U y6 2⁄----------------------=

Ue

LvK

#k--

# 1cµ

1 4⁄--------- k-----=

-

6%-t6-----------

6U j%-x j6-----------------+ &%S2 '%-2 6

x j6-------µt,-------6-x j6-------2 34 5+–=

+ 2%,#------- 1

-----k6x j6-------6-

x j6-------k-2------

6-x j6-------6-

x j6-------–2 34 5 )2

˜ *%S2 LLvK--------+

+ %-k-------

6x j6-------

vt,------- k6

x j6-------2 34 5 1

,k------ 1

,#-------–2 3

4 5



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 toactivate the SAS capability in URANS regions. In RANS regime (and particularly in boundarylayers) 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.

)2˜ )2 cSAS:= L k cµ

0.25-( )⁄=

2%,#------- 1

-----k6x j6-------6-

x j6-------

k-" k--

k-- k-#

FSST-SAS2%,#------- k

-2------6-

x j6-------6-

x j6-------– )2˜ *%S2 L

LvK--------+=

LvK

2%,#------- k

-2------6-

x j6-------6-

x j6------- )2˜ *%S2 L

LvK--------d (RANS regime)

2%,#------- k

-2------6-

x j6-------6-

x j6------- )2˜ *%S2 L

LvK--------< (SAS regime)

FSST

1k2-----

6kx j6-------

6kx j6-------

)2˜

FSST-SAS =

% FSASmax )2˜ *S2 L

LvK-------- 2

,#-------k max 1

-2------6-

x j6-------6-

x j6-------1k2-----

6kx j6-------

6kx j6-------,2 3

4 5:– 0,:

k-#

FSAS 1.25= )2˜ )2 cSAS: 1.755= = ,#, , 2 3⁄=

FSAS

FSST-SAS -

Turbulence and Wall Function Theory: Modeling Flow Near the Wall


It should be noted that the details of the formulation of the second derivative term areimportant. 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 thegrid limit.

Similar to the DES formulation, the SAS model also benefits from a switch in the numericaltreatment between the steady and the unsteady regions. In DES, this is achieved by ablending function as proposed by Strelets [58], which allows the use of a second orderupwind scheme with the ANSYS CFX-Solver in RANS regions and a second order centralscheme 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 factthat no RANS-shielding is performed for the SAS-SST model. For details, see Discretizationof 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 modeledin ANSYS CFX. An introduction to near-wall flow, modeling details and guidelines on usingwall functions are presented. For details, see Modeling Flow Near the Wall (p. 125 in "ANSYSCFX-Solver Modeling Guide").

Mathematical Formulation

The wall-function approach in ANSYS CFX is an extension of the method of Launder andSpalding [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 byemploying empirical formulas to provide near-wall boundary conditions for the mean flowand turbulence transport equations. These formulas connect the wall conditions (e.g., thewall-shear-stress) to the dependent variables at the near-wall mesh node which ispresumed to lie in the fully-turbulent region of the boundary layer.

The logarithmic relation for the near wall velocity is given by:

(Eqn. 164)

Ue62U i

6x j2-----------62U i

6x j2-----------= 6S

6x j-------- 6S

6x j--------f

.-

u+ U tu.------ 1

*--- y+( )ln C+= =



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 wallvelocity, 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+ %Byu.µ----------------=

u..-%-----2 3

4 51 2⁄

=

u.

By .-*

By

u* Cµ1 4⁄ k1 2⁄=

u.

u.U t

1*--- y*( )ln C+------------------------------=

.-

.- %u*u.=

y* %u*By( ) µ⁄=



One of the major drawbacks of the wall-function approach is that the predictions dependon 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 caseof fine meshes, can be overcome with the use of the Scalable Wall Function formulationdeveloped by ANSYS CFX. It can be applied on arbitrarily fine meshes and allows you toperform a consistent mesh refinement independent of the Reynolds number of theapplication.

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 sublayerand 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 greaterthan 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 toensure a sufficient number of nodes in the boundary layer.

If the results deviate greatly from these ranges, the mesh at the designated Wall boundarieswill require modification, unless wall shear stress and heat transfer are not important in thesimulation.

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 canbe one of three different treatments in ANSYS CFX. They are based on different distancedefinitions and velocity scales. This has partly historic reasons, but is mainly motivated bythe 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+ .- %⁄ Bn:v------------------------------=

Bn

y+

By Bn/4=



• 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 onarbitrarily fine meshes.

AutomaticNear-WallTreatment forOmega-BasedModels

While the wall-functions presented above allow for a consistent mesh refinement, they arebased 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 massand momentum balance. For flows at low Reynolds numbers, this can cause an error in thedisplacement thickness of up to 25%. It is therefore desirable to offer a formulation whichwill 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 tobe zero and the flux in the momentum equation is computed from the velocity profile. Theequations are as follows:

Flux for the momentum equation, FU:

(Eqn. 173)

with:

(Eqn. 174)

(Eqn. 175)

Flux for the k-equation:

(Eqn. 176)

By Bn/4=

By Bn=

y+ max y* 11.06( , )= y* u*Bn 4⁄v--------------------=

k---

-

FU %u. u*–=

u. v BUBy--------

=

u* max a1k u.( , )=

Fk 0=



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 followingformulation is selected:

(Eqn. 179)

While in the wall-function formulation, the first point is treated as being outside the edge ofthe viscous sublayer, the location of the first mesh point is now virtually moved downthrough the viscous sublayer as the mesh is refined in the low-Re mode. It is to beemphasized, that the physical location of the first mesh point is always at the wall (y = 0). Theerror in the wall-function formulation results from this virtual shift, which amounts to areduction 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 ashydraulically 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].

--

-lu*

a1*y------------ 1a1*v------------ u*2

y+--------= =

-s6v

' By( )2-----------------=

By

-- -s 1-l-s-----2 3

4 5 2+=

B

u+ 1*---

y*1 0.3k++----------------------2 34 5ln C+=

k+ yR%µ---u*=



You are advised to exercise care in the use of the rough wall option together with thestandard wall-function approach. Inaccuracies can arise if an equivalent sand grainroughness 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 "ANSYSCFX-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 walltreatment. Using similar assumptions as those above, the non-dimensional near-walltemperature 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+ %cpu* Tw T– f( )qw

----------------------------------------=

T+ Pr y*e $–( ) 2.12 y*( )ln '+[ ]e 1– $⁄( )+=

' 3.85Pr1 3⁄ 1.3–( )2

2.12 Pr( )ln+=

$ 0.01 Pr y*( )4

1 5Pr3 y*+-----------------------------=

Prµcp+--------=

+

qw%cpu*

T+--------------- Tw T f–( )=



Turbulent fluid flow and heat transfer problems without conjugate heat transfer objectsrequire 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 bythe surface area and adding to the corresponding boundary control volume energyequation. If the wall temperature is specified, the wall heat flux is computed from theequation above, multiplied by the surface area and added to the boundary energy controlvolume equation.

AdditionalVariables

The treatment of additional scalar variables in the near wall region is similar to that for heatflux.

Treatment ofCompressibilityEffects

With increasing Mach number (Ma > 3), the accuracy of the wall-functions approachdegrades, which can result in substantial errors in predicted shear stress, wall heat transferand wall temperature for supersonic flows.

It has been found that the incompressible law-of-the-wall is also applicable to compressibleflows 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 followingequation (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 compu.

--------------- 1*---

u* yv---------2 3

4 5ln C+=

u. .w %w⁄( )1 2⁄= y+ u. y vw⁄( )= *

U comp%%w------ Ud7=

. .w=

q qw U .w+=

. µtU6y6-------=



and

(Eqn. 193)

If (Eqn. 190), (Eqn. 192) and (Eqn. 193) are substituted into (Eqn. 191) and integrated, theresulting equation is:

(Eqn. 194)

which provides a relationship between the temperature and velocity profiles. Using theperfect 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 thefollowing 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 ofwall-functions which are applicable to compressible flows. These wall-functions areprimarily implemented in two locations in ANSYS CFX: hydrodynamics and energy. First,consider the implementation in the hydrodynamics section. The equation for thewall-shear-stress is obtained after a slight rearrangement of (Eqn. 188):

(Eqn. 199)

. µtU6y6-------=

T TwPrt qwU

cp .w--------------------–

Prt U 2

2cp----------------–=

U comp B A U+D---------------2 3

4 5 AD----2 3

4 5asin–asin=

A qw .w⁄=

B 2cpTw( ) Prt( )⁄=

D A2 B+=

.w%-----

u*1k--

u* yv---------ln C+

-----------------------------U comp=



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 isgiven 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*------------------2 3

4 5 B*D*-------2 3

4 5asin–asin=

u* Cµ1 4⁄ k1 2⁄=

.*w % u*( )2=

Ec U 2

cp T---------=

B*qw

cp T .*w-----------------U=

D* B*2 2Prt-------

TwT-------Ec+2 3

4 5 1 2⁄=

U comp cpT Ec( )comp( )1 2⁄=

C 5.2=Prt 0.9=

TwT------- 1 Prt B*

Prt2------- Ec+ +=

\+

\+ PrtUu*-----=

\+ PrtUu*-------

%cp u*qw

---------------- Tw T–Prt U 2

2cp---------------–

2 38 94 5

= =



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 walltemperature is known, then the wall heat transfer is calculated). Before proceeding, it isinstructive to consider the equation that is used in ANSYS CFX for low Mach number heattransfer. The equation is subsequently modified for use at higher Mach numbers. Thisunmodified 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 wallregions, 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 byinterpreting the linear and logarithmic velocity profiles given above as “compressible” orVan 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)

\+

\+ Pr U 1+( ) e $– Prt U 2

+( ) e 1 $⁄–+=

U 1+ y+=

U 2+ 1

Prt------- 2.12 y+( )ln '+( )=

$ f 1 Pr y+( )=

' f 2 Pr( )=

U comp1+ y+=

U comp2+ 1

Prt------- 2.12 y+( )ln '+( )=

U + 1R--- RU comp

+( )sin H 1 RU comp+( )cos–( )–=

Turbulence and Wall Function Theory: Wall Distance Formulation


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-walland 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*( )⁄=

\+

wall

y

y=o

0

y2

2

dd 0 1–=

0 0=

ydd0 y– C1+=

0 0.5 y2– C1 y C2+ +=

Turbulence and Wall Function Theory: Wall Distance Formulation


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.

0 0= y 0= C2 0=

C1 ydd0 y+=

C1 C2 y

0yd

d0

0 0.5 y2– ydd0 y+2 34 5 y+=

a y2 by c+ + 0=

a 0.5= b ydd0= c 0–=

y ydd0– yd

d02 34 5

220++=

0C2 1–=

0 0=

ydd0 0C yd

d0

y

Wall Distance C0– C0 2 20++=

0



GGI and MFR Theory

Introduction


• Interface Characteristics (p. 120)

• Numerics (p. 120)

This chapter provides an overview of the numerical methods used for GGI (general gridinterface) in ANSYS CFX.

A control surface approach is used to perform the connection across a GGI attachment orperiodic condition. A physically based intersection algorithm is employed to provide thecomplete freedom to change the grid topology and physical distribution across theinterface.

A general intersection algorithm permits connections to be successfully made, even whenthe resultant surfaces on either side of an interface do not physically “fit” together to form awell defined physical connection. In addition, an automatic surface trimming function isperformed by the GGI algorithm, to account for mismatched surface extent. This means thata GGI attachment or periodic condition can be successfully defined where the surface onone side of the interface is larger (in extent) than the surface on the other side of theinterface. The interface is constructed between the overlapping regions of the two sides ofthe interface.

Multiple Frames of Reference (MFR) allows the analysis of situations involving domains thatare rotating relative to one another. For ANSYS CFX, this feature focuses on the investigationof rotor/stator interaction for rotating machinery using the Frozen Rotor frame changemodel. Since MFR is based on the GGI technology, the most appropriate meshing style maybe used for each component in the analysis.

GGI and MFR Theory: Interface Characteristics


Interface Characteristics

The numerical algorithms employed, as well as the control surface treatment of thenumerical fluxes across the interface, are designed and implemented in such a way as toprovide for maximum robustness and accuracy. The treatment of the interface fluxes is fullyimplicit and fully conservative in mass, momentum, energy, scalars, etc. This means that themultigrid solver can be applied directly, without any penalty in terms of robustness orconvergence rate, to problems involving GGI conditions. Each of the different types of GGIinterfaces 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 nodaldependent variables, and in terms of control surface equations and control surfacevariables.

Numerics

If the case is transient rotor-stator, then the current relative position of each side of a slidinginterface is first computed at the start of each timestep. Each frame change model thenproceeds by discretizing the surface fluxes along each side of the interface in terms of nodaldependent variables, and in terms of control surface equations and control surfacevariables. Each interface surface flow is discretized using the standard flux discretizationapproach, but employs both the nodal dependent variables on the local side of the interfaceand control surface variables on the interface region. Balance equations within the interfaceregion are generated for the interface variables from the flux contributions from both sidesof the interface. These equations are called control surface equations (different from controlvolume equations) because they enforce a balance of surface flows over a given surfacearea.

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


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 usingthe regular shape function based gradient coefficients, but all dependence of thegradient estimate on nodal pressure on the interface is in terms of the interfacepressure 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 èxposed 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 tolocal interface variables as well as to nodal variables.

5. Solve the linear equation set to yield values of all nodal variables (from the controlvolume equations) and all interface variables (from the control surface equations).

GGI and MFR Theory: Numerics




Multiphase Flow Theory

Introduction


• 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–Eulerianmultiphase model and a Lagrangian Particle Tracking multiphase model. Additionalinformation on the Lagrangian Particle Tracking model theory is available. For details, seeParticle Transport Theory (p. 187).

This section covers the Eulerian-Eulerianmodel theory. For details, see Multiphase FlowModeling (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


Multiphase Notation

In addition to the notation given here, review the list of symbols. For details, see List ofSymbols (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 singlephase flows treat total pressure differently depending on the simulation. For details, seeTotal Pressure (p. 14).

& ' 1& ' 1

& r& V&

& V r&

V& r&V=

NP r&& 1 to NP=

& %&

& &

%& r&%&=

& &&

%m %&r&&V=

ptot pstat12--r&

%&U&2

&V+=

Multiphase Flow Theory: The Homogeneous and Inhomogeneous Models


The Homogeneous and Inhomogeneous Models

Two different sub-models are available for Eulerian-Eulerian multiphase flow, thehomogeneous 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 contactsurface 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. Theseessentially 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)

& ' A&'

& ''

d'

A&'6r'd'--------=

r'

r' '

r'

A&'6r'`d'-----------=

r'`max r' rmin,( ) if r' rmax<( )

max1 r'–

1 rmax–-------------------rmax rmin,( ) if r' rmax>( );]=]?

=



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 particleReynolds number and the fluid Prandtl number. These are defined using the particlemean 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 behaviorin these two limits is given by:

(Eqn. 11)

rmax rmin

rmin

rmin

Re&'%& U' U&– d'

µ&-------------------------------------=

Pr&'µ&CP&+&

----------------=

µ& CP& +&

&

& '

A&'r&r'd&'----------=

d&'

r& 0b r' 0b

A&'6r&r'

r&d' r'd&+-----------------------------= d&'r&d' r'd&+

6-----------------------------=g



Non-dimensional interphase transfer coefficients may be correlated in terms of the mixtureReynolds 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 justtwo 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 asother relevant fields such as temperature and turbulence. This allows some simplificationsto be made to the multifluid model resulting in the homogeneous model.

For a given transport process, the homogeneous model assumes that the transportedquantities (with the exception of volume fraction) for that process are the same for allphases, i.e.,

(Eqn. 17)

Since transported quantities are shared in homogeneous multiphase flow, it is sufficient tosolve for the shared fields using bulk transport equations rather than solving individualphasic transport equations.

Re&'%&' U' U&– d&'

µ&'------------------------------------------=

Pr&'µCP&'+&'----------------=

%&' µ&' CP&' +&'

%&' r&%& r'%'+=

µ&' r&µ& r'µ'+=

A&' r&C=

A&'2 rC & rC '

rC & rC '+-------------------------------=

0& 0= 1 & NPh h

Multiphase Flow Theory: Hydrodynamic Equations


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. Forexample, the velocity field may be modeled as inhomogeneous, but coupled with ahomogeneous turbulence model. Alternatively, a homogeneous velocity field may becoupled with inhomogeneous temperature fields. Homogeneous additional variables arealso available in ANSYS CFX.

Hydrodynamic Equations

The following is a summary of the equations of momentum and mass transfer forinhomogeneous and homogeneous multiphase flow in ANSYS CFX. The equivalent forsingle-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).

0

t66 %0( ) %U0 $ 0C–( )C•+ S=

% r&%&& 1=

N P

V=

U 1%--- r&%&U&& 1=

N P

V=

$ r&$&& 1=

N P

V=

t66 r&%&U&( ) r& %&U& U&D( )( )C•+

r& p&C– r&µ& U&C U&C( )T+( )( )C•+=

$&'+ U ' $'&

+ U&–( )' 1=

N p

V SM& M&+ + +

SM&

M& &



• 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 solidphases, additional terms are present representing additional stresses due to particlecollisions.

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 atransported volume conservation equation. Take (Eqn. 22), divide by phasic density, andsum over all phases. This yields:

(Eqn. 24)

Interpreting this equation is simpler if you consider the special case of incompressiblephases with no sources, in which it simplifies to:

(Eqn. 25)

which requires the volume flows to have zero divergence. (Eqn. 24) is the volume continuityequation solved by the ANSYS CFX-Solver.

$&'+ U ' $'&

+ U&–( )

66t----- r&%&( ) C r&%&U&( )•+ SMS& $&'

' 1=

N p

V+=

SMS&

$&' ' &

r&& 1=

N P

V 1=

1%&------

%6 &t6--------- C r&%&U&( )•+2 3

4 5&V 1

%&------ SMS& $&'

' 1=

N P

V+2 38 94 5

&V=

C r&U&( )•&V 0=



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 samepressure 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 homogeneousmultiphase flow.

4NP 1+ 5NP

U& V& W& r& p& NP 1–

p& p for all & 1,......,NP= =

U& U= 1 & NPh h,

t66 %U( ) %U UD µ UC UC( )T+( )–( )C•+ SM pC–=

% r&%&& 1=

N P

V=

µ r&µ&& 1=

N P

V=

U

U

Multiphase Flow Theory: Multicomponent Multiphase Flow


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 multicomponentmass 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. Whenusing the homogeneous model, momentum transfer between phases is assumed to be verylarge.

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 netinterfacial forces sum to zero:

(Eqn. 32)

The total interfacial force acting between two phases may arise from several independentphysical effects:

(Eqn. 33)

The forces indicated above respectively represent the interphase drag force, lift force, walllubrication force, virtual mass force, turbulence dispersion force and solids pressure force(for dense solid particle phases only).

Y A&

t66 r&%&Y A&( ) r& %&U&Y A& %&DA& Y A&C( )–( )( )C•+ SA&=

%&DA& DA&

M&'

& ' &M&

M& M&'' &fV=

M&' M– '&=( ) M&&V 0=g

M&' M&'D M&'

L M&'LUB M&'

VM M&'TD MS ....+ + + + + +=

Multiphase Flow Theory: Interphase Momentum Transfer Models


ANSYS CFX provides a wide range of physical models for these forces. These models aredescribed 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 alsodescribed.

The total drag force is most conveniently expressed in terms of the dimensionless dragcoefficient:

(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 .

&'

M& c&'d( ) U ' U&–( )=

c&& 0= c&' c'&=

c&'d( )

CDD

12--%& U& U'–( )2A------------------------------------------=

% U& U'–( ) D

A& '



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.

c&'d( )

Ap V p

ApGd2

4---------=

V pGd3

6---------=

d np

npr'V p-------

6r'Gd3---------= =

Dp12--CD%&Ap U ' U&– U ' U&–( )=

D&' npDp34--

CDd-------r'%& U ' U&– U ' U&–( )= =

&

D&' c&'d( ) U ' U&–( )=

c&'d( ) 3

4--CDd-------r'%& U ' U&–=

c&'d( ) CD

8-------A&'%& U ' U&–=



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 pastspherical 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 todominate viscous effects (the inertial or Newton’s regime), the drag coefficient becomesindependent 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 dragcoefficient is a complex function of Reynolds number, which must be determined fromexperiment.

This has been done in detail for spherical particles. Several empirical correlations areavailable. The one available in ANSYS CFX is due to Schiller and Naumann (1933) [6]. It canbe written as follows:

Schiller Naumann Drag Model

(Eqn. 45)

ANSYS CFX modifies this to ensure the correct limiting behavior in the inertial regime bytaking:

(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 amodified 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–h h

0.1 Re 1000< <

CD24Re------ 1 0.15Re0.687+( )=

CD max 24Re------ 1 0.15Re0.687+( ) 0.44,2 34 5=

CD rc1.65– max 24

ReW-------- 1 0.15Re'0.687+( ) 0.44,2 34 5=

Re' rcRe=

rc



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 avoidsubsequent numerical difficulties, ANSYS CFX modifies the original Gidaspow model bylinearly 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 thecorrelation 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 behavein the same manner as solid spherical particles. Hence, the drag coefficient is wellapproximated by the Schiller-Naumann correlation described above.

At larger particle Reynolds numbers, the inertial or distorted particle regime, surface tensioneffects become important. The fluid particles become, at first, approximately ellipsoidal inshape, 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 IshiiZuber 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 thedimensionless group known as the Eotvos number, which measures the ratio betweengravitational 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>

c&'d( ) 150

1 rc–( )2µc

rcd p2-------------------------- 7

4--1 rc–( )%c U c U d–

d p---------------------------------------------- ,+= rc 0.8<

rd 0.2<

0.7 rc 0.8< <

CD cap( ) 83--=

EogB%d p

2

,----------------=

B% g,



The Ishii-Zuber correlation gives:

(Eqn. 51)

In this case, ANSYS CFX automatically takes into account the spherical particle and sphericalcap limits by setting:

(Eqn. 52)

The Ishii Zuber Model also automatically takes into account dense fluid particle effects. Fordetails, 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--

gdU T

2-------B%%c-------=

U T

U Tµc

%cd p-----------M 0.149– J 0.857–( )=

Mµc

4 gB%

%2,3---------------- Morton Number= =

J0.94H0.751 2 H 59.3h<

3.42H0.441 H 59.3>284

=

H 43--EoM 0.149– µc

µref--------2 34 5 0.14–

=

µref 0.0009 kg m 1– s 1–=



In this case, ANSYS CFX automatically takes into account the spherical particle and sphericalcap 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 donein 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 amixture 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 multiplyingfactor 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+( )=

Rem%c U d U c– d p

µm----------------------------------=

µmµc------- 1

rdrdm--------–2 3

4 5 2.5rdmµ*–=

µ*µd 0.4µc+

µd µc+-------------------------=

rdm

CD ellipse( ) E rd( )CDF=

CDF23--Eo1 2⁄=

E rd( )1 17.67 f rd( )6 7⁄+( )

18.67 f rd( )------------------------------------------------=

f rd( )µcµm------- 1 rd–( )1 2⁄=



Densely Distributed Fluid Particles: Dense Spherical Cap Regime (Ishii Zuber)

(Eqn. 61)

The Ishii Zuber correlation, as implemented in ANSYS CFX, automatically selects flow regimeas 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 SparselyDistributed 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–( )2CDF=

CDF83--=

CD CD sphere( )= if CD sphere( ) CD ellipse( )i

CD min CD ellipse( ) CD cap( ),( )= if CD sphere( ) CD ellipse( )<

CD rcpCDF=

CDF

CD

D&' CD%&'A&' U' U&– U' U&–( )=

D&' ' &

%&'

%&' r&%& r'%'+=

A&'

A&'r&r'd&'----------=



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 mixturemodel (see “Interphase Drag for the Mixture Model (p. 138)”), except that the interfacial areadensity 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 inthe 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 aconstant, or a CEL expression.

Virtual Mass Force

The virtual mass force is proportional to relative phasic accelerations as follows:

(Eqn. 71)

d&'

A&' r&C=

A&'2 rC & rC '

rC & rC '+-------------------------------=

FcL F– d

L rd%cCL U d U c–( ) -cA= =

-c curl U c=

CL

^

FcL F– d

L rd%cCL U d* U c

*–( ) -c* 2^+( )A= =

-c* curl U c

*=

FcVM F– d

VM rd%cCVMDdU d

Dt--------------DcU c

Dt-------------–2 34 5= =



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 dispersedphase is observed to concentrate in a region close to the wall, but not immediately adjacentto the wall. This effect may be modeled by the wall lubrication force, which tends to pushthe 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 canonly be expected on extremely fine meshes.

Note that these default coefficients are rather different from those recommended in theoriginal paper of Antal et al [88]. More recent numerical experiments by Krepper and Prasser

[89] obtained the values ,

^ U *

FcVM F– d

VM rd%cCVMDd

*U d*

Dt--------------Dc

*U c*

Dt-------------– 2^ U d* U c

*–( )A+2 38 94 5

= =

CVM 0.5=

CVM CVM

FcLUB F– d

LUB r– d%cU r U r nw:( )nw–( )( )2

d p------------------------------------------------------max C1 C2

d pyw------+ 0,2 3

4 5nw= =

U r U c U d–= d p

yw nw

C1 0.01–= C2 0.05=

ywd p------ C1

C2------ 5dh

C1 0.0064–= C2 0.016=

Multiphase Flow Theory: Solid Particle Collision Models


Interphase Turbulent Dispersion Force

Favre AveragedDrag Model

ANSYS CFX implements a model for turbulent dispersion force, based on the Favre averageof 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 turbulentdispersion 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 oforder 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 modelsimplemented 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 CTDCcdEtc,tc-------

rdCrd---------

rcCrc--------–2 3

4 5= =

Ccd

,tc

0.9

CTD

McTD M– d

TD CTD%ckc rcC–= =

CTD CTD

CTD



Solids Stress Tensor

Additional stresses due to inter-particle collisions are modeled using a collisional solidsstress 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 arespecified using empirical constitutive relations. See, for example, (Enwald et al [[97]]). Inmany of these, the solids pressure, shear and bulk viscosities are simple functions of thesolid phase volume fraction.

Kinetic TheoryModels

These are a class of models, based on the kinetic theory of gases, generalized to take intoaccount inelastic particle collisions. In these models, the constitutive elements of the solidsstress are functions of the solid phase granular temperature, defined to be proportional tothe 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 atransport equation. However, in many circumstances, it is possible to ignore the transportterms, 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.

.sij Ps(ij– µsU i6x j6---------

U j6xi6---------

23--–

U k6xk6---------(ij+2 3

4 5 )sU k6xk6---------(ij+ +=

Ps µs )s

!s13--u's

2=

Ps Ps rs( ) CPsg G rs( )Crs= =

G rs( ) G0ec rs rsm–( )=

G rs( ) G0 c

rsm



The Gidaspow model is implemented with an option for specifying the Reference ElasticityModulus 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 afunction of volume fraction only.


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 givenby:

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 ispossible for the user to specify a model for this, using a CEL expression, if required. Note thatany solids volume fraction dependence must be included in the CEL expression.

G rs( )

Ps %srs!s 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 rsmb

g0 rs( ) C0 C1 rs rc–( ) C2 rs rc–( )2 C3 rs rc–( )3+ + += r rci( ),

rc rsm 0.001–=

C0 1079=

C1 1.08 106A=

C2 1.08 109A=

C3 1.08 1012A=




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 Millerand Gidaspow (1992) [106]. They successfully modeled gas-solid flow in a riser using a solidsshear viscosity linearly proportional to the solids phase volume fraction.

(Eqn. 87)

Note that their constant of proportionality is dimensional, and is likely to requiremodification for different fluid-solid material properties.

More complex models for solids shear stress allow the shear stress to become very large inthe limit of maximum packing. A wide range of such models is discussed in the review articleby Enwald et al [97].

It is possible for the user to implement any of these models, using a CEL expression for thesolids shear viscosity. Note that any solids volume fraction dependence must be included inthe CEL expression.


Typically, the shear viscosity is expressed as a sum of at least two contributions: the kineticand collisional contributions:

(Eqn. 88)

There is wide agreement on the correct form of the collisional contribution. As in the kinetictheory 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 kineticcontribution. For example:

Gidaspow (1994) [18]:

(Eqn. 90)

)s43--rs

2%sd p g0 1 e+( ) !G----=

µs 5.35 Poise[ ] rA s=

µs µs col, µs kin,+=

µs col,45--rs

2%sd p g0 1 e+( ) !G----=

µs kin,5 G48----------

%sd p1 e+( )g0---------------------- 1 4

5-- 1 e+( )g0rs+2 34 5 2

!=



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 G96----------%sd p

1Tg0--------- 8

5--rs+2 34 5

1 85--T 3T 2–( )rs g0+

2 T–-----------------------------------------------2 38 98 94 5

!=

T 12-- 1 e+( )=

!s

!s

Production Dissipation .sijU i6x j6---------g 1 s= =

.sij

1 s 3 1 e2–( )rs2%s g0!s

4ds----

!sG------

U k6xk6---------–

2 38 94 5

2 38 94 5

=

.sijU i6x j6--------- Ps

U k6xk6---------– µs

U i6x j6---------

U j6xi6---------+2 3

4 5 U i6x j6--------- )s

23--µs–2 3

4 5 U k6xk6---------2 3

4 5 2+ +=

Pg roduction PsD– µsS2 +sD

2+ +=

+s )s23--µs–= D

U k6xk6---------= S2 1

2--U i6x j6---------

U j6xi6---------+2 3

4 5 2=



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 granulartemperature 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)

!s

!s

Ps !sa µs )s !s1 2⁄a,

Ps Ps0( )!s= µs µs

0( )!s1 2⁄=

)s )s0( )!s

1 2⁄= +s +s0( )!s

1 2⁄=

Production +s0( )D2 µs

0( )S2+( )!s1 2⁄ Ps

0( )D!s–=

AP!s1 2⁄ BP!s–=

AP +s0( )D2 µs

0( )S2 0i+= BP Ps0( )D=

Dissipation AD!s3 2⁄ BD!s–=

AD4

ds G------------CD 0i=

BD CDD=

CD 3 1 e2–( )rs2%s g0 0i=

!s1 2⁄ !s

AD!s BP BD–( )!s1 2⁄ AP–+ 0=

Multiphase Flow Theory: Interphase Heat Transfer


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 temperaturescan be generated in regions of low solid particle volume fraction. To circumvent this, it isrecommended that you specify an upper bound for the granular temperature. The squareof 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 singlephase energy equations (Eqn. 6) and (Eqn. 7).

The multiphase version of the Total Energy equation is:

(Eqn. 107)

AD

AP 0i

!s1 2⁄ BD BP– BD BP–( )2 4ADAP++

2AD--------------------------------------------------------------------------------=

!s1

15 1 e–( )----------------------ds2S2=

t66 r&%&h& tot,( ) r&

p6t6------ r&%&U&h& tot, r&+&CT&–( )C•+–

r&µ& CU& CU&( )T 23--CU&(–+2 3

4 5U&2 34 5C•–

$&'+ h' tot, $'a

+ h& tot,–( )' 1=

N p

V Q& S&+ +=



The multiphase version of the thermal energy equation for static enthalpy (incompressibleand 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 InterphaseMass 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)

t66 r&%h&( ) r& %&U&h& +& T&C–( )( )C•+

$&'+ h's $'a

+ h&s–( )' 1=

N p

V Q& S&+ +=

h& T& +&

&

S&

Q& &

$&'+ h's $'a

+ h&s–( )

&Q&

Q& Q&'' &fV=

Q&' Q– '&= Q&&V 0=g



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 dimensionlessNusselt 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

.

h&'

Q&' A&'

' &

Q&' h&'A&' T' T&–( )=

Q&' c&'h( ) T' T&–( )=

c&'h( )

c&'h( ) h&'A&'=

h&'

h +Nud-----------=

+d

h&'+&Nu&'

d'--------------------=

Nu 2=Re

Pr µ&CP& +&⁄=



Additional information on models in ANSYS CFX is available. For details, see Particle ModelCorrelations for Overall Heat Transfer Coefficient (p. 172 in "ANSYS CFX-Solver ModelingGuide"). Some additional details for the Interface Flux model are provided below.

• Interface FluxThe heat flux coefficients for both fluids and the interfacial heat flux value, F12, fromFluid 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 phase1 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 bulktemperature difference my be recovered using:

Mixture ModelCorrelations

When using the mixture model, the Nusselt number is defined in terms of a mixtureconductivity 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 notsufficient to model the interphase heat transfer process. A more general class of modelsconsiders separate heat transfer processes either side of the phase interface. This is achievedby using two heat transfer coefficients defined on each side of the phase interface.

Q21 Q12– A12F12= =

F12

F12

h16F126T1----------- 0i ,d h2

6F216T2-----------d 0i

F12 F21– h T1 T2–( ),= = h1 h2 h= =

h&'+&'Nu&'

d&'---------------------- ,= +&' r&+& r'+'+=



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 MixtureModel).

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 somewhatsuperfluous, as it may be implemented using a user-specified overall heat transfercoefficient.

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 HeatTransfer Coefficients (p. 173 in "ANSYS CFX-Solver Modeling Guide").

&

q& h& Ts T&–( )=

'

q' h' Ts T'–( )=

h& h' & ' Ts

Nu&h&d&'+&

---------------=

+& & d&'

q& q'+ 0=

Tsh&T& h'T'+

h& h'+--------------------------------=

q& q'– h&' T' T&–( ),= = 1h&'-------- 1

h&------ 1

h'-----+=

h& Fb

Ts T&=

Multiphase Flow Theory: Multiple Size Group (MUSIG) Model


Homogeneous Heat Transfer in Multiphase Flow

For all transport processes other than heat transfer, the shared field is the same as thetransported quantity in the equation. However, in the case of heat transfer, it is temperaturewhich is shared but enthalpy which is transported. Hence, ANSYS CFX does not solve a bulkenthalpy equation, but rather solves a separate enthalpy equation for each fluid with a largeinterphase heat transfer term which forces the temperature fields to be the same.

The equations solved are identical to the phasic equations for full multiphase describedabove. For homogeneous heat transfer model, the interphase heat transfer coefficient is notmodeled by any of the correlations used in full multiphase. Instead it is chosen to be largerelative to the other transported processes in the equation, thereby ensuring the phasictemperatures are the same.

Multiple Size Group (MUSIG) Model

The MUSIG (Multiple Size Group) model has been developed to handle polydispersedmultiphase flows. By polydispersed, it means that the dispersed phase has a large variationin size. One of the attributes of polydispersed multiphase flow is that the different sizes ofthe dispersed phases interact with each other through the mechanisms of breakup andcoalescence. Population balance is a well-established method for calculating the sizedistribution of a polydispersed phase, including breakup and coalescence effects. MUSIGprovides a framework in which the population balance method can be incorporated intothree-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

t66 n v t,( )

xi66 U i v t,( )n v t,( )( )+ BB DB– BC DC–+=



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 tocoalescence 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 " v;( )n " t,( ) "dvF7=

DB n v t,( ) g v ";( ) "d0v7=

BC12-- Q v " ";–( )n v " t,–( )n v t,( ) "d0

v7=

DC n v t,( ) Q v ";( )n " t,( ) td0F7=

g v ";( ) v" v "– Q v ";( )

v "v "+

Ni

i

Ni t( ) n v t,( ) vdvi 1 2⁄–

vi 1 2⁄+7=

i mi ri

miNi %iri=

mi

miNi t( )( )6t6---------------------------

xi66 U i

i t( )Ni t( )( )+ Si=

%iri( )6t6---------------- xi6

6 %iriU ii( )+ Si=



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⁄=

t66 %ird f i( )

xi66 %irdU i

i f i( )+ Si=

%d udi

t66 %drd f i( )

xi66 %drdU d

i f i( )+ Si=

BBi mi BB vdvi 1 2⁄–

vi 1 2⁄+7=

mi g v j vi;( )N jj i>V=

%drd g v j vi;( ) f jj i>V2 3

4 5=

DBi %drd f i g vi v j;( )j 1<V2 3

4 5=

BBi DBi–( )iV 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;]]]=]]]?

=



The contribution of the birth rate due to coalescence of smaller particles to the source termin (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 sourceto 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 andmaximum diameter for the polydispersed fluid must be set along with the number of sizegroups. 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 thisexpression to ensure that the group sizes are independent of local density.

BCi mi BC vdvi 1 2⁄–

vi 1 2⁄+7=

mi12-- Q v j vk;( )X jkiN jNk

k ihV

j ihV2 3

4 5=

%drd( )2 12-- Q v j vk;( )X jki f j f k

m j mk+m jmk--------------------

k ihV

j ihV2 3

4 5=

DCi %drd( )2 Q vi v j;( ) f i f j1

m j------

jV2 34 5=

BCi DCi–( )iV 0=

Q vi v j;( ) Q vd vi;( )=

j k,

X jkiiV 1=

m G6---%d

d3=



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 distributionresolves the small bubbles well, and the equal diameter distribution weights both equally.

i

mi mmin Bm i 12--–2 3

4 5+=

Bmmmax mmin–

N--------------------------------=

i

di dmin Bd i 12--–2 3

4 5+=

Bddmax dmin–

N----------------------------=

mi mmin34--

mmax mmin–

2N i–--------------------------------+=

Group Equal Mass Equal Diameter Geometric Mass1 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=



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 bubblesin turbulent suspensions. The model is based on the theory of isotropic turbulence andprobability. 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–( )"c

di2-----2 3

8 94 5 1 3⁄ 1 j+( )2

j11 3⁄-------------------e Z– jdjmin

17=

Z12 f BV

2 3⁄ 1 f BV–( )2 3⁄ 1–+( ),

'%c"c2 3⁄ di

5 3⁄ j11 3⁄------------------------------------------------------------------------=

j

jmin 11.4Tdi----=

T 1"c----Ec

32 34 5 1 4⁄

=



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 inthree steps. First, the bubbles collide trapping a small amount of liquid between them. Thisliquid film then drains until the liquid film separating the bubbles reaches a criticalthickness. The film them ruptures and the bubbles join together.

The coalescence kernel is therefore modeled by a collision rate of two bubbles and acollision 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 ' 2= "c

Ec ,

g vi v j;( )

i j

Q vi v j;( ) \ijT \ij

B \ijS+ +( )Tij=

tij

.ij

Tij etij .ij⁄–=

tij%crij

3

16,----------2 38 94 5

1 2⁄

lnh0h f-----2 34 5=

.ijrij

2 3⁄

"c1 3⁄------------=

h0 h f

rij

rij12--

1ri--- 1

r j----+2 3

4 52 34 5 1–

=

Multiphase Flow Theory: The Algebraic Slip Model


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 ANSYSCFX.

\ijT FCT Sij uti

2 utj2+( )

1 2⁄=

SijG4--- di d j+( )2=

uti 2"c1 3⁄ di

1 3⁄=

FCT

\ijB FCBSij U rj U ri–=

U ri2.14,%cdi------------- 0.505g di+=

FCB

Q vi v j;( )

ij

Q vi v j;( ) Q v j vi;( )=



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)

&

%&r&( )6t6-------------------

%&r&u&i( )6

xi6--------------------------+ 0=

%&r&u&i( )6

t6--------------------------%&r&u&

j u&i( )6

x j6---------------------------------+ r&

p6xi6-------–

r&.&ji( )6

x j6------------------- r&%&g i M&

i+ + +=

M&i

%m6t6----------

%mumi( )6

xi6----------------------+ 0=

%mumi( )6

t6----------------------%mum

j umi( )6

x j6-----------------------------+ p6

xi6-------–

.mji .D

ji+( )6

x j6-------------------------- %m gi+ +=

%m r&%&&V=

%mumi r&%&u&

i

&V=

.m r&.&&V=

.Dji r&%& u&

i umi–( )u&

j

&V–=



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 massfraction and drift velocity as:

(Eqn. 172)

Derivation of the Algebraic Slip Equation

The phasic and bulk momentum equations are first transformed to nonconservative formby combining with the phasic and bulk continuity equations. The phasic momentumequation then becomes:

(Eqn. 173)

and the bulk momentum equation becomes:

(Eqn. 174)

uS&i u&

i uci–=

uD&i u&

i umi–=

uD&i uS&

i Y&uS&i

&V–=

%mY&( )6t6----------------------

xi66 %mY& um

i uD&i+( )( )+ 0=

%&r&u&

i6t6--------- %&r&u&

j u&i6

x j6---------+ r&

p6xi6-------–

r&.&ji( )6

x j6------------------- r&%&g i M&

i+ + +=

%mum

i6t6--------- %mum

j umi6

x j6---------+ p6

xi6-------–

.mji .D

ji+( )6

x j6-------------------------- %m gi+ +=



(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 andthat 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 automaticallyincluded by taking the derivative of the absolute frame velocity rather than relative framevelocity on the right-hand-side.

M&i r& %&

uD&i6t6------------ %& %m–( )

umi6t6---------+

2 38 94 5

=

r& %&u&j u&

i6

x j6--------- %mum

j umi6

x j6---------–

2 38 94 5

+

r&.&ji( )6

x j6------------------- r&

.mji .D

ji+( )6

x j6--------------------------+–

r& %& %m–( )gi–

u&j u&

i6

x j6--------- um

j umi6

x j6---------d

M&i r& %& %m–( )

umi6t6--------- um

j umi6

x j6--------- g i–+

2 38 94 5

=

M&i 3

4--–r&%cd p-----------C

DuS& uS&

i=

uS& uS&i 4

3--–d p

%cCD------------ %& %m–( )

umi6t6--------- um

j umi6

x j6--------- g i–+

2 38 94 5

=

Multiphase Flow Theory: Turbulence Modeling in Multiphase Flow


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 turbulenceeffects do not affect the continuity equation; instead, they appear as an apparent turbulentdispersion force in the momentum equation.

In the ASM formulation outlined here, turbulent dispersion forces are not considered in thederivation of slip velocity. Instead, turbulent dispersion is modeled using the sameturbulence model as for multicomponent flows:

(Eqn. 180)

Energy Equation

For multicomponent fluids, the energy equation has a additional term corresponding toenthalpy transport by species velocities. For standard transported components, this velocityis 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 thecalculation. A sink for the ASM mass fraction is defined accordingly. The implicit assumptionis that this mass is replaced by an equal mass of the ballast (constraint) component. Theenergy equation also removes the enthalpy of the ASM species and adds the enthalpy of theballast species.

Turbulence Modeling in Multiphase Flow

This section describes the extension of the single-phase turbulence models to multiphasesimulations. 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).

.Dji

%&Y&eu&i e

µtPrt-------

Y&6

xi6----------=

x j66 %mY&uD&

j h&&

N c

V–



Phase-Dependent Turbulence Models

Phase dependent turbulence models can be used in conjunction with the inhomogeneousmodel (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 themean 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.

&

µ&eff µ& µt&+=

U t&

lt&

µt& %& f µU t&lt&=

f µ

&

lt&V D

1 3⁄

7---------------=

V D

EtdEtc,------= µtdg

%d%c-----

µtc,-------=

,Etd Etc

, 1=

, 1>



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 phaseversion, with all flux and volumetric source terms multiplied by volume fractions. Singlephase 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 usersources.

Turbulence Enhancement

Sato [22] successfully modeled Particle Induced Turbulence this for bubbly flow using anenhanced continuous phase eddy viscosity:

(Eqn. 189)

k "

µt& cµ%&k&

2

"&--------2 38 94 5

=

k "

t66 r&%&k&( ) r& %&U&k& µ

µt&,k--------+2 3

4 5 k&C–2 34 5

2 34 5C•+

r& P& %&"&–( ) T&'k( )+=

t66 r&%&"&( ) r&%&U&"& µ

µt&,"--------+2 3

4 5 "&C–2 34 5C•+

r&"&k&----- C"1P& C"2%&"&–( ) T&'

"( )+=

T&'k( ) T&'

"( ) k "

µtc µts µtp+=

Multiphase Flow Theory: Additional Variables in Multiphase Flow


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 sameas the single phase equations, except that the mixture density and the mixture viscosity areused. Single phase turbulence model information is available. For details, see TurbulenceModels (p. 69).

For inhomogeneous multiphase flow, it is possible to solve a single turbulence field in asimilar way to homogeneous flow. For details, see Homogeneous Turbulence inInhomogeneous 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 Cµp%crdd p U d U c–=

Cµp

# & #&

t66 r&%&0&( ) r&%&U&0&( )C• r& %&D&

0( ) µt&Sct&----------+2 3

4 5 0&C2 34 5C•–+

S&#( ) T&

#( )+=

0& &

#& & #& %&0&=

D&#( ) &

S&#( ) &

T&#( ) 0&

Sct& &



As in single phase, Diffusive Transport and Poisson equations are also available formultiphase additional variables). For details, see Additional Variables (p. 40). Diffusivetransport equations exclude the advection term, turbulent diffusion term, andcontributions due to interphase mass transfer from (Eqn. 191). Poisson equations excludethese 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 phaseinterface:

(Eqn. 194)

(Eqn. 195)

The first of these is used if the additional variable is defined per unit mass. The latter is usedif 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)

#&

M' & '

#&

T&#( ) T&'

#( )

' &fV=

T&'#( ) T– &'

#( )= T&#( )

&Vg 0=

#& M'

T&'#( ) c&'

0( ) U' 0&–( )=

T&'#( ) c&'

#( ) M' #&–( )=

c&'#( )

.&' #&

#&

T&'#( ) .&'A&' U' 0&–( ),= for specific variables

T&'#( ) .&'A&' M' #&–( ),= for volumetric variables



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.

c&'#( ) .&'A&'=

Sh

. $Shd----------=

$ D%D

$d

.&'$&Sh&'

d'------------------=

Sh 2=Re

Pr µ& %D( )&⁄=

F12 #1

M2

T21 T12– A12F12= =

F12



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 mixturediffusivity 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

c16F126#1----------- 0i ,d c2

6F216M2-----------d 0,i if volumetric

c16F12601----------- 0i ,d c2

6F216U2-----------d 0,i if specific

F12 F21– c 01 U2–( ),= = c1 c2 c= =

.&'$&'Sh&'

d&'--------------------- ,= $&' r&$& r'$'+=

#& # 1 & N ph h=

t66 %0( ) %U0 $ 0C–( )C•+ S=

% r&%&& 1=

N P

V= U 1%--- r&%&U&& 1=

N P

V= $ r&$&& 1=

N P

V=, ,

Multiphase Flow Theory: Sources in Multiphase Flow


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 additionalvariables. For this reason, only specific additional variables are permitted to behomogeneous in ANSYS CFX.

Sources in Multiphase Flow


• 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 areimplemented on a phase basis. For details, see Sources (p. 43). You may need to multiply amultiphase 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 thisproperty. They are applied at a fluid-independent level, and are therefore automaticallyadded 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 isapplicable 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).

66t----- r&%&( ) C r&%&U&( )•+ S& $&+=

S&

Multiphase Flow Theory: Interphase Mass Transfer


• 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 volumetricmass 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 automaticlinearization 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 inducesadditional sources and sinks between all other transport equations. These are referred to assecondary sources, or secondary fluxes, due to mass transfer.

The default form of the secondary source terms for a transported variable is:

(Eqn. 213)

$& &

$& $&'' 1=

N p

V=

$&' ' &

$&' $'&–= $&& 1=

N p

V 0=g

$&'

$&' $&'+ $'&

+–=

$&'+ 0> '

&

$&' m&'A&'=

m&' ' & A&'

$&'

0&

SM& $&'+ 0' $'&

+ 0&–( )' 1=

N p

V=



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 ofdiscontinuities in the transported variable at the phase interface, for example, enthalpy inthe 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 transfersources.

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 sourcesof 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)

' & 0'&

m&'

$&' m&'A&'=

$&'

SM&' $&'+ 0' $'&

+ 0&–( )=

&'

cA& Molar Concentration of Component A in phase &=

%A& Mass Concentration of Component A in phase &=

XA& Mole Fraction of Component A in phase &=

Y A& Mass Fraction of Component A in phase &=



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, theconcentrations of A in each phase are not the same. However, there is a well definedequilibrium curve relating the two concentrations at equilibrium. This is often, though notalways, expressed in terms of mole fractions:

(Eqn. 218)

For binary mixtures, the equilibrium curve depends on temperature and pressure. Formulticomponent mixtures, it also depends on mixture composition. The equilibrium curveis in general monotonic and non-linear. Nevertheless, it is convenient to quasi-linearize theequilibrium 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 ofthe 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, dynamicequilibrium occurs when the partial pressure of the vapor A is equal to its saturated vaporpressure at the same temperature.

%A& MAcA&=

XA&cA&c&--------=

Y A&%A&%&---------=

& '

XA& f XA'( )=

XA& K A&'x XA'= K A&'

x mole fraction equilibrium ratio=

Y A& K A&'y Y A'= K A&'

y mass fraction equilibrium ratio=

cA& K A&'c cA'= K A&'

c molar concentration equilibrium ratio=

%A& K A&'% %A'= K A&'

% mass concentration equilibrium ratio=

K A&'% K A&'

c c&c'-----K A&'

x %&%'------K A&'

y= = =



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 needsto be generalized. Henry’s law states that a linear relationship exists between the molefraction 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 dissolvedin 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 molefraction 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 molarconcentration 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=



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 transfercoefficient. Instead, it is necessary to consider a generalization of the Two Resistance Modelpreviously 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 equilibriumconditions prevail at the phase interface.

Also, it is assumed in this section that the total mass transfer due to species transfer issufficiently small that primary mass sources to the phasic continuity equations areneglected. Hence, secondary mass fluxes to the species mass transfer equations are alsoignored.

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:

& '

cA'S

cA&S

cAS

cA'

Phase & Phase 'Interface

Molar Concentration of A

k&c k'

c

&

nA& k&c cA&s cA&–( )=

'

nA' k'c cA's cA'–( )=

&

mA& k&c %A&s %A&–( )=



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) todetermine the interface mass concentrations:

(Eqn. 234)

These may be used to eliminate the interface values in (Eqn. 230) and (Eqn. 231) in order toexpress 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 theflow. Hence, it may be used to simulate evaporation and condensation, or melting andsolidification. For example, it may be used to model condensation of saturated vaporbubbles in sub-cooled liquid, or evaporation of saturated bubbles in super-heated liquid.

Modeling advice and consideration information is available. For details, see Thermal PhaseChange 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 conjunctionwith 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)

'

mA' k'c %A's %A'–( )=

%&s %'s

cA&s K ccA's= %A&s Kc%A's=g

mA& mA'+ 0=

%A&s

Kc----------- %A'sk&

c %A& k'c %A'+

K ck&c k'

c+-------------------------------------= =

mA& mA'– k&'c Kc%A' %A&–( ),= = 1

k&'c--------

1k&

c-----Kc

k'c------+=

&

q& h& Ts T&–( )=



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 thetotal 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 discontinuityin static enthalpy due to latent heat between the two phases. This is achieved using amodification 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 upwindformulation. However, the saturation enthalpy is carried into the incoming phase. Thus:

(Eqn. 241)

'

q' h' Ts T'–( )=

h& h' & '

Ts T sat=

q& q'+ 0=

&

Q& q& m&'H&s+=

'

Q' q' m&'H's–=

m&' & '

H&s H's

Q& Q'+ 0=

m&'q&' q'&+H's H&s–------------------------=

m&' 0> H&s H&sat, H's H'==g

m&' 0< H&s H&, H's H'sat==g



This leads to a formulation which is stable both physically and numerically. It implies thatthe 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 tendencyfor a flow to cavitate increases as the cavitation number is decreased.

Cavitation is treated separately from Thermal Phase Change, as the cavitation process istypically 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 isdirected towards models which take both effects into account.

In ANSYS CFX, the Rayleigh Plesset model is implemented in the multiphase framework asan 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 vaporgeneration and condensation. The Rayleigh-Plesset equation describing the growth of a gasbubble 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 thermalbarriers to bubble growth. Neglecting the second order terms (which is appropriate for lowoscillation frequencies) and the surface tension, this equation reduces to:

(Eqn. 244)

L H'sat H&sat–=

Cap pv–12--%U 2--------------=

p pv

RBd2RB

dt2------------ 32--

dRBdt---------2 3

4 52 2,

% f RB------------+ +

pv p–% f

--------------=

RB pv

p% f ,

dRBdt---------

23--

pv p–% f

--------------=



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 begeneralized 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 isusually much slower than vaporization).

Despite the fact that (Eqn. 249) has been generalized for vaporization and condensation, itrequires further modification in the case of vaporization.

Vaporization is initiated at nucleation sites (most commonly non-condensible gases). As thevapor 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---------- d

dt-----43--GRB

32 34 5 4GRB

2 23--

pv p–% f

--------------= =

dmBdt----------- %g

dV Bdt---------- 4GRB

2 %g23--

pv p–% f

--------------= =

NB rg

rg V BNB43--GRB

3 NB= =

m fg NBdmB

dt-----------3rg %g

RB---------------- 2

3--pv p–% f

--------------= =

m fg F3rg%g

RB-------------- 2

3--pv p–% f

------------------- pv p–( )sgn=

F

rg rnuc 1 rg–( )

m fg F3rnuc 1 rg–( )%g

RB-------------------------------------- 2

3--pv p–% f

------------------- pv p–( )sgn=



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 bubbleconcentration and radius are required. The Rayleigh Plesset cavitation model implementedin 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 UserDefined Cavitation Models (p. 185 in "ANSYS CFX-Solver Modeling Guide").

When using a user defined cavitation model, the ANSYS CFX-Solver will perform genericlinearizations for the volume fraction and volume continuity equations to help stability andconvergence. The saturation pressure is used by the ANSYS CFX-Solver in linearizing thecavitation 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 subsequentdroplet condensation. It is also useful to model additional condensation when droplets arealready present in significant quantities. Typical applications include low-pressure steamturbines, 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 orappear through various nucleation mechanisms, including homogeneous (volumetric) andheterogeneous (surface) nucleation. In the current release, only homogeneous nucleationmechanisms have been considered.

The Droplet Condensation Model differs from the Thermal Phase Change model (see TheThermal Phase Change Model (p. 176)) in that the droplet diameter is calculated as part ofthe model rather than as a user input. This leads to improved accuracy. In order to do this atransport equation for droplet number must also be solved, which has as its source termnucleation contributions. This model differs from the Equilibrium Phase Change model (seeEquilibrium Phase Change Model (p. 314 in "ANSYS CFX-Solver Modeling Guide")) in that itdoes not assume the flow to instantaneously reach equilibrium conditions, and thereforeimplicitly includes losses due to thermodynamic irreversibility.

In the following discussion, we consider a homogenous multiphase system, in which thedroplets move with the same velocity as the continuous phase. However, the model hasbeen generalized to inhomogeneous systems as well.

rnuc

RB

RB 1 µm=

rnuc 5 E 4–=

Fvap 50=

Fcond 0.01=



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 continuousphase, 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 fractionswhere:

(Eqn. 254)

Global continuity and momentum equations are also solved as described in HomogeneousHydrodynamic Equations (p. 130). The continuous phase energy equation, in total enthalpyform, is

(Eqn. 255)

6%crc6t------------- 6

6xi------- %cuirc( )+ Sd m*rcJd+( )

i 1=

nd

V–=

nd

d

6%drd6t-------------- 6

6xi------- %duird( )+ Sd m*rcJd+=

6%cNd6t---------------- 6

6xi------- %cuiNd( )+ %drcJd=

Jd

m*

r*

rc rdi 1=

nd

V+ 1=

6%crcHc6t-------------------- 6

6xi------- %crcuiHc( )+ rc–( )6P

6t------66xi------- $trc

6T6xi-------2 3

4 5 SH+ +=



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 largedroplets, 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 nucleatedradius (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 conjunctionwith 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 classicalnucleation 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 propertydatabase must be used that evaluates supercooled state properties. This requires anequation of state for the vapor phase amenable to extrapolation into regions within thesaturation 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---------

kcRd%d 1 cKn+( )-------------------------------------

Td Tg–hg hp–------------------2 3

4 5=

J A BG*

kTg----------–

2 38 94 5

exp=

A G*

k Tg



The interaction between the phases by mass transfer depends on calculating the dropletdiameter, 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 growthrate 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 thedroplet surface and its internal temperature). This implies that almost all of the heat transfereither comes from the continuous phase during evaporation or goes into it duringcondensation.

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

4GNd--------------2 34 5

13--

=

'd3rdRd--------=

Sd %d'dRddtd---------=

SH SdHu– 'dQd+=

Hu

Qd

Qdkc

Rd 1 cKn+( )------------------------------ Td Tg–( )=

µm

NuKn

Nu 21 cKn+( )------------------------=

Multiphase Flow Theory: Free Surface Flow


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 ofhomogeneous 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 keepthe 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 flowremain well behaved at the interface.

Surface Tension

The surface tension model used in ANSYS CFX is based on the Continuum Surface Forcemodel of Brackbill et al [27]. This models the surface tension force as a volume forceconcentrated at the interface, rather than a surface force. Consider the free surface interfaceshown 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, '

Primary Fluid, &

n

\

& '

F&' f &'(&'=



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 andtangential components of the surface tension force respectively. The normal componentarises from the interface curvature and the tangential component from variations in thesurface 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 walladhesion by specifying the contact angle which the interface makes with the wall throughthe primary fluid. The interface normal vector used for the calculations of both curvatureand the surface tension force must satisfy the wall contact angle.

f &' ,&' *&' n&'– Cs,+=

(&' Cr&'=

, n&'

CS *

*&' n&'C•=

(&'





Particle Transport Theory

Introduction


• 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 trackedthrough 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 iscarried 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. Theseequations are then integrated using a simple integration method to calculate the behaviorof the particles as they traverse the flow domain. The following section describes themethodology used to track the particles

Lagrangian Tracking Implementation

Within the particle transport model, the total flow of the particle phase is modeled bytracking a small number of particles through the continuum fluid. The particles could besolid particles, drops or bubbles.

The application of Lagrangian tracking in ANSYS CFX involves the integration of particlepaths through the discretized domain. Individual particles are tracked from their injectionpoint until they escape the domain or some integration limit criterion is met. Each particleis injected, in turn, to obtain an average of all particle tracks and to generate source termsto the fluid mass, momentum and energy equations. Because each particle is tracked fromits injection point to final destination, the tracking procedure is applicable to steady stateflow analysis. The following section describes the methodology used to track the particles.

Particle Transport Theory: Lagrangian Tracking Implementation


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 thetimestep is assumed to prevail over the entire step. At the end of the timestep, the newparticle 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, f0would 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 velocityare needed at the position of the particle. These variables are always obtained accurately bycalculating the element in which the particle is traveling, calculating the computationalposition within the element, and using the underlying shape functions of the discretizationalgorithm 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 anda difference in velocity between the particle and fluid. Conversely, there is a counteractinginfluence of the particle on the fluid flow due to the viscous drag. This effect is termedcoupling between the phases. If the fluid is allowed to influence trajectories but particles donot affect the fluid, then the interaction is termed one-way coupling. If the particles alsoaffect the fluid behavior, then the interaction is termed two-way coupling.

The flow prediction of the two phases in one-way coupled systems is relativelystraightforward. The fluid flow field may be calculated irrespective of the particletrajectories. One-way coupling may be an acceptable approximation in flows with lowdispersed 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 momentumequations. The momentum sources could be due to turbulent dispersion forces or drag. Theparticle 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.

(t

xin xi

o vpio (t+=

vpi

vp v f vpo v f–( ) (t

.-----–2 34 5exp .Fall 1 (t

.-----–2 34 5exp–2 3

4 5+ +=

Particle Transport Theory: Momentum Transfer


The particle sources to the momentum equations are obtained by solving transportequations 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 flowrate 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. Thesource terms are then retained in memory in order that they may be applied each time thefluid coefficients are calculated. Thus, the particle sources may be applied even thoughparticles 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 theparticle which affect the particle acceleration are due to the difference in velocity betweenthe particle and fluid, as well as to the displacement of the fluid by the particle. The equationof motion for such a particle was derived by Basset, Boussinesq and Oseen for a rotatingreference 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--------- CS0P RS+=

CS0P

RS

mPUPdtd---------- FD FB FR FVM FP FBA+ + + + +=

FD

FB

FR

FVM

FP

FBA



The left hand side of Equation 4 can be modified due to the special form of the virtual massterm (see Virtual or Added Mass Force (p. 192)) which leads to the following form of theparticle 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 termcorrection) 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 FWVM FP+ + +( ) 1

mP-------FR+=

FWVM

mPG6---dP

3%P= and mFG6---dP

3%F=

dP %F %P

RVMmP

mPCVM

2-----------mF+---------------------------------

%P

%PCVM

2-----------%F+------------------------------= =

1 RVM–

CVM2-----------%F

%PCVM

2-----------%F+------------------------------=

RVM

UPdtd----------

RVMmP----------- FD FB FWVM FP+ + +( ) 1

mP-------FR+=

UP

T R ClinUP+=

RClin



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-Eulerianmultiphase 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 equalto 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--CD%FAF U S U S

12--CD%FAF UF UP– UF UP–( )= =

CD AF

CD

D 12--CD%FAF U S=

R D

mPCVM

2-----------mF+---------------------------------UF

RVMmP-----------DUF= =

ClinD

mPCVM

2-----------mF+---------------------------------–

RVMmP-----------D–= =

FB mP mF–( )g mP 1%F%P------–2 3

4 5 g G6---dP

3 %P %F–( )g= = =

g

R%P %F–

%PCVM

2-----------%F+------------------------------g RVM 1

%F%P------–2 3

4 5 g= =



Rotation Force

In a rotating frame of reference, the rotation term is an intrinsic part of the acceleration inand is the sum of Coriolis and centripetal forces:

(Eqn. 15)

As described elsewhere (see Pressure Gradient Force (p. 193) and Virtual or Added MassForce (p. 192)), the implemented rotation term also contains contributions from thepressure gradient and the virtual mass force due to the domain rotation which leads to thefollowing 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 surroundingfluid, leading to an additional drag which has the following form:

(Eqn. 18)

FR mP 2^ UPA– ^ ^ rPAA–( )=

R 2^ UPA– ^ ^ rPAA–=

1 RVM–( ) 2^ UFA– ^ ^ rPAA–( )– (virtual mass force)

RVM%F%P------– 2^ UFA– ^ ^ rPAA–( ) (pressure gradient term)

RCor,P 2^ UPA–=

RCor,F 2^ UFA–=

RCent ^ ^ rPAA–=

R RCor,P C 1–( )RCor,F CRCent+ +=

C 1=

C RVM=

C 1 %F %P⁄–=

C RVM 1 %F %P⁄–( )=

FVMCVM

2-----------mFUFdtd----------

UPdtd----------–2 3

4 5=



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 (seeEquation 5), leading to a modified effective mass. Considering only steady state flows, theremaining 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 particleand is defined as:

(Eqn. 23)

This force is only important if large fluids pressure gradients exist and if the particle densityis smaller than or similar to the fluid density. Neglecting diffusive and source terms in thesteady state momentum equation, the pressure gradient can be replaced by the velocitygradient. Assuming a constant fluids density, the pressure gradient force can be written as

(Eqn. 24)

CVM

CVM RVM

FWVMCVM

2-----------mF UFCUF RF–( )=

R 1 RVM–( ) UFCUF RF–( )=

RF 2^ UFA– ^ ^ rPAA–=

RF

R 1 RVM–( ) UFCUF( )=

FPmF%F-------– pC=

FP mF UFCUF RF–( ) mP%F%P------ UFCUF RF–( )= =



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 separatetrajectories due to the random nature of the instantaneous fluid velocity. It is the fluctuatingcomponent 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 theparticle 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 RVM%F%P------ UFCUF RF–( )=

RF 2^ UFA– ^ ^ rPAA–=

RF

R RVM%F%P------UFCUF=

v f

v f

v f

v f

v f W

v f W

.e

v f W

v f W

.e

Particle Transport Theory: Heat and Mass Transfer


The turbulent velocity, eddy and length and lifetime are calculated based on the localturbulence 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 ofparticles (usually an order of magnitude higher) since a stochastic method is used. This willgreatly increase computational time; therefore, this is most often performed as apost-process where there is only one particle iteration.

If turbulent dispersion is used in an iterative situation, it may not be possible to achievecomplete convergence because of the stochastic nature of the sources to the continuousphase, although the random number generator used in determining the eddies is reset oneach particle iteration.

Heat and Mass Transfer


• 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 W $ 2k 3⁄( )0.5=

leCµ

3 4⁄ k3 2⁄

"----------------------=

.e le 2k 3⁄( )1 2⁄⁄=

" Cµ

Cµ3 4⁄

$

uW vW wW, ,( )



Heat Transfer

The rate of change of temperature is governed by three physical processes: convective heattransfer, 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 takingplace. The latent heat of vaporization V is temperature dependent, and is obtained directlyfrom 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 thosenot affected by mass transfer.

QC Gd+Nu TG T–( )=

+

Nu 2 0.6Re0.5 µCP+------2 3

4 513--

+=

QMdmC

dt-----------VV=

QR

"p

Qr14--"pGd p

2 I ,nT p4–( )=

G,

mCCP( )V tddT QC QM QR+ +=



Simple Mass Transfer

Each component of mass being transferred between the continuous and particle phasessatisfies 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 concentrationdifferences. While this may be appropriate in some situations, such as solids having a smallmoisture content, it does not adequately account for the vapor pressure dependence onparticle temperature, which is imported for evaporating liquids. In these situations, theliquid evaporation model, presented below, is more appropriate.

Liquid Evaporation Model

The liquid evaporation model is a model for particles with heat transfer and one componentof mass transfer, and in which the continuous gas phase is at a higher temperature than theparticles. The model uses two mass transfer correlations depending on whether the dropletis above or below the boiling point. This is determined through the Antoine equation and isgiven 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 theconvective heat transfer:

(Eqn. 38)

dmCdt----------- Gd%DSh E m: F mFG–( )–=

mC mF mFG

E%D Sh

Sh 2 0.6Re0.5 µ%D-------2 34 5

13--

+=

Pvap Pref A BT C+--------------–2 3

4 5exp=

tddm QC

V-------–=



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 hasevaporated.

Oil Evaporation/Combustion

The oil combustion model uses the particle transport model to track evaporating oildroplets, which are modeled in a very similar way to the liquid evaporation model, and usesthe 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, theNusselt number and the Sherwood number on the gas assumed to be in the boundary layerof 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 boundarylayer consists entirely of the local gas mixture.

Reactions

Arrhenius reactions can be set up in a particle calculation. Reactants must just be in theparticle, but the products can be both in the particle and in the continuous phase. The heatof 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 coalparticles with an eddy dissipation calculation for the combustion of the volatile gases in thegas phase. Two separate gases are given off by the particles, the volatiles and the charproducts which come from the burning of carbon within the particle.

Gas phase combustion is modeled by means of regular single phase reactions. A transportequation is solved in the fluid for each material given off by the particles. The ‘volatiles’ maybe 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 internalreactions, heat transfer and full coupling of mass, momentum and energy with the gaseousphase. The combustion of a coal particle is a two stage process: the devolatilization of theraw coal particle followed by the oxidation of the residual char to leave incombustible ash.

tddm GdDSh

WCWG--------- 1 X–

1 XG–----------------2 34 5log=



The devolatilization is usually modeled as a single step or a two step process. The charoxidation is modeled either as a global reaction, or using an analytical solution for thediffusion and reaction of oxygen within the pores of the char particle.

The devolatilization and char oxidation processes can occur on time scales of ordermilliseconds which are several orders of magnitude less than the typical residence time ofthe particle in the furnace. Large variations in time scales can result in numerically stiffequations, which can cause accuracy problems with explicit integration algorithms. TheANSYS CFX implementation of the particle transport model bases its timestep on thereaction rate to ensure that the solution has the required accuracy.

Devolatilization Devolatilization is modeled using the generic Arrhenius reactions capability. Twoalternative devolatilization models are also used for coal combustion. The simpler model isthe single reaction model of Badzioch and Hawksley [78]. At time t, assume that a coalparticle, 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 tworeactions 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 proximateanalysis, where the heating rate is low and the volatiles escaping from the bed of coal mayundergo secondary reactions including cracking and carbon deposition on solid surfaces. Ithas been found experimentally that the potential yield of volatiles from pulverized coalparticles 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------–2 3

4 5exp=



produce a yield greater by as much as a factor of two than the proximate value. If the singlereaction model is used, it is difficult to know which data to use, since the coal particlesexperience a wide range of temperatures as they disperse in the furnace.

Bituminous coals generally have a volatile yield, which depends strongly on temperatureand heating rates. In such cases, it is important to take account of this dependence, forexample, by using a multiple reaction model of devolatilization. As an alternative to thesingle reaction model, it is better to use the model of Ubhayakar et al. [79] in which tworeactions with different rate parameters and volatile yields compete to pyrolyse the rawcoal. 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, andwill 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 volatilessince 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. Thefractional increase in the mean radius of the coal particles due to devolatilization is requiredas 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=



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 inputparameters.

• 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 overallchar 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-------------------2 34 5 &PA

P------=

&

kc AcTCTP-------–2 3

4 5exp=

kd1– kc

1–+( )1–molfrcO2

4GRp2 PPA------

0

0C O2+ 2 0 1–( )CO 2 0–( )CO2+b



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 forthe 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. Thecoefficient 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)

0

2 0 1–( )2 0–-------------------- AS

TST p------–2 3

4 5exp=

tddmc 30

1 "–-----------–Mc

MO2-----------

%F%c------ k1

1– k2 k3+( ) 1–+( )1–mc=

k1DR2-----=

DDref%fluid-----------

T p T g+2Tref-------------------2 34 5=

&

k2 1 "–( )kcR----=

kc AcT pTcT p------–2 3

4 5exp=



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 zoneto the raw coal as it emerges from the burner. In general, this pre-heating is supplied by acombination of convective heating by recirculating product gases, as well as by theabsorption 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 alinear 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 solidor liquid hydrocarbon fuel in a user-friendly way. The solver uses the provided informationto derive the parameters of related objects, e.g., initial particle mass fractions, materialproperties 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 standardanalysis 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 andchlorine)

k3 kc ' 'coth 1–( ) '2a( )⁄=

' Rkc

Dp"a-------------2 34 5 0.5

=

Dp effic DA=

"p

"p 1 f v–( )"p coal( ) f v"p char( )+=



Some additional input data is required by the solver, for which the default values should beappropriate 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 oxidisesulphur 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 actualvolatiles yield plus the moisture. Carbon, hydrogen and oxygen content of the volatiles arecomputed from ultimate analysis, which in turn defines the stoichiometric coefficients inthe gas phase reactions involving the volatiles material. When the fuel nitrogen model isenabled, corrections are made in order to account for the carbon and hydrogen released asHCN.

Particle Transport Theory: Basic Erosion Model


Basic Erosion Model


• 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 ofparticle impact, particle and wall properties. For nearly all metals, erosion is found to varywith 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)


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 1( )=

E V P f 1( )

n

n 2=

E kV P2 f 1( )=

f 1( ) 13--cos21 if 1tan 1

3-->=

f 1( ) sin(21 ) 3sin21 if 1tan 13--h–=

k

EV PV 0-------2 34 5 n

f 1( )=

V 01kn

-------2 34 5



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.


The model constants in the original formulation of Tabakoff and Grant are only valid forparticle velocities specified in feet per second [ft/s]. The Tabakoff formulation is modified inANSYS CFX as outlined below:

(Eqn. 70)

E

E k1 f 1( )V P2 cos21 1 RT

2–[ ] f V PN( )+=

f 1( ) 1 k2k12 1 G 2⁄1 0----------2 3

4 5sin+2

=

RT 1 k4V Psin1–=

f V PN( ) k3 V P 1sin( )4=

k21.0 if 1 21 0h

0.0 if 1 21 0>;=?

=

EV P 1

1 0 k1 k4

k12 1 0

E f 1( )V PV 1-------2 34 5 2

cos21 1 RT2–[ ] f V PN( )+=



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 adimension of velocity. The following list shows the link between the constants of theoriginal 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



[deg] Angle of Maximum Erosion

f 1( ) 1 k2k12 1 G 2⁄1 0----------2 3

4 5sin+2

=

RT 1V PV 3-------sin1–=

f V PN( )V PV 2------- 1sin2 34 5 4

=

k21.0 if 1 21 0h

0.0 if 1 21 0>;=?

=

k12

k2

V 1

V 2

V 3

1 0

V 1 1 k1⁄=

V 2 1 k34( )⁄=

V 3 1 k4⁄=

Particle Transport Theory: Spray Breakup Models


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 qualitativeguide 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 industrialapplications. Examples are spray painting or fuel injection systems. The following sectionsgive 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 conditionsfor 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 andturbulence 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 acomprehensive 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 ofdroplets is the so-called "blob-method." In this approach, it is assumed that a detaileddescription 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=



Assuming non-cavitating flow inside the nozzle, it is possible to compute the dropletinjection 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 inANSYS CFX.

Figure 1 Blob-method

Enhanced BlobMethod

Kuensberg et al [110] have suggested an enhanced version of the blob-method. Similar tothe blob-method, it is assumed that the atomization processes need not be resolved indetail. Contrary to the standard blob-method this method allows to calculate an effectiveinjection velocity and an effective injection particle diameter taking into account thereduction of the nozzle cross section due to cavitation.

During the injection process, the model determines if the flow inside the nozzle is cavitatingor not and dynamically changes the injection particle diameter and the particle injectionvelocity. 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( )Anozzle%P------------------------=

Anozzle mnozzle

Pvena Pvapor

Pvena t( ) P1%P2------U vena

2 t( )–=

U vena t( )U mean t( )

Cc----------------------=

Cc



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( )Anozzle%P-----------------------=

Dnozzle

Pvena Pvapor

U eff Deff

U eff t( )Anozzlem t( )---------------- Pvapor P2–( ) U vena t( )+=

Deff t( )4Aeff t( )

G--------------------=

Aeff t( ) m U eff t( )%P( )⁄=



• Vapor pressure of the particle fluid.

• Normal distance of the pressure probe from the injection center. The fluid pressure atthis position (marked as position 2 in Figure 2) will be used to determine theacceleration 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 effectsof the kind of primary break-up that is described below and presented in detail by Senecalet al [126].

In direct injection spark ignition engines, pressure swirl atomizers are often used in order toestablish hollow cone sprays. These sprays are typically characterized by high atomizationefficiencies. With pressure swirl injectors, the fuel is set into a rotational motion and theresulting 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, thetangential motion of the fuel is transformed into a radial component and a liquid sheet isformed. This sheet is subject to aerodynamic instabilities that cause it to break up intoligaments.

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 G%uh0 d0 h0–( )=



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 oninternal 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 ofvelocity is assumed to remain constant.

Sheet Breakup and AtomizationAfter the liquid film has left the injector nozzle, it is subject to aerodynamic instabilities thatcause it to break-up into ligaments. The theoretical development of the model is given indetail 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 pB%l----------=

kv

kv max 0.7 4md0

2%l \cos----------------------

%l2 pB----------,=

pBu

u U !cos=

! h0

w U !sin=

2h U

T T0eikx -t+=

T0 k 2G( ) +⁄=

- -r i-i+= ^

-r



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)

^

-r2vlk

2 kh( )tanhkh( )tanh Q+------------------------------------–=

4vl2k4 kh( )tanh( )2 Q2U 2k2– kh( )tanh Q+( ) ,k3

%l--------- QU 2k2–2 38 94 5

–

kh( )tanh Q+-------------------------------------------------------------------------------------------------------------------------------------------------------------+

Q %g %l⁄= ^ L .

L U----TbT0------2 34 5ln=

. 1----TbT0------2 34 5ln=

Tb T0⁄( )ln

dl

W eg 27 16⁄<

dl

dl8hK s------=

Ks ^

dl

dl16hK s---------=

Ks

Ks%gU 2

2,-------------=



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 isneeded:

• 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 froma homogeneous binary mixture. It is also possible to specify the material vapor pressuredirectly by using the option Vapor Pressure.

Primary Breakup Model RestrictionsPrimary break-up models are used to determine starting conditions for the droplets thatleave the injection nozzle. To accomplish this task, the models use averaged nozzle outletquantities as input for the calculation of initial droplet diameters, break-up times anddroplet 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 chosento compute the particle material vapor pressure inside the nozzle.

Secondary Breakup Models

The following sections give an overview about the currently available secondary breakupmodels in ANSYS CFX. Further information about the breakup models can be found in theprovided references.

d0 1.88dl 1 3Oh+( )1 6⁄=

Oh

Oh WeRe------------=



BreakupRegimes

The breakup of a liquid jet into droplets is caused by a combination of differentmechanisms: turbulence within the liquid phase, implosion of cavitation bubbles andexternal aerodynamic forces acting on the liquid jet. Depending on the injection parameterssuch as the relative velocity between liquid and gas, the liquid and gas densities and theliquid viscosity and surface tension the contribution of each of the above mechanisms to thespray 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 ofunity. Above a certain value of the Weber number, the droplet deformation leads tobreakup. Typically, the following breakup regimes are observed [117]:

Numericalapproach tobreakupmodeling

For the numerical simulation of droplet breakup, a so-called statistical breakup approachedis used in ANSYS CFX. In this framework, it is assumed that if a droplet breaks up into childdroplets, the particle diameter is decreased accordingly to the predictions of the usedbreakup 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 singlerepresentative particle.

Reitz andDiwakarBreakup Model

This model ([115]) distinguishes between two breakup regimes: bag breakup and strippingbreakup. Breakup occurs if a critical particle Weber number has been exceeded.Independent of the breakup regime, it is assumed that during breakup the followingrelation 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 <

We%FV slip

2 rP,-----------------------=

On µ%P,DP---------------------=

WeWeWeWeWe

L

tddrP rP rstable–( )–

tbr------------------------------=

rstable

tbr rstable



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 onexperimental findings of Hsinag et al. [109] and Pilch et al. [117]. It can be shown thatirrespective of the breakup regime, the time to deform a particle from a sphere into a diskshape 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 C1%PrP

3

2,-----------=

rstable6,

%FV slip2-----------------=

We Re⁄ Cs1>

tbr C2r

V slip-----------

%P%P------=

rstable,2

2%F2 V slip

3 E------------------------=

C1 C2 W ecrit Cs1

C1 G

C2

W ecrit

Cs1

ti 1.6*t*=



with the characteristic time t*

(Eqn. 105)

The second phase of breakup, which is characterized by further distortion of the droplet toits final destruction is modeled by the following correlations:

(Eqn. 106)

Please note that the last two breakup regimes are an extension to the originally proposedmodel 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-----------

%P%F------=

tbrt*------

6 We 12–( ) 0.25– 12 We 18<h

2.45 We 12–( )0.25 18 We 45<h

14.1 We 12–( ) 0.25– 45 We 351<h

0.766 We 12–( )0.25 351 We 2670<h5.5 2670 Weh;

]]]=]]]?

=

tbrt*------ 4.5 1*1.2On0.74( )=

Bt ti– tbr= Bt

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



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 andis 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 Tayloranalogy. Within the Taylor analogy, it is assumed that the droplet distortion can bedescribed as a one-dimensional, forced, damped, harmonic oscillation similar to the one ofa 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 viscosityacts as a damping force and the surface tension as a restoring force, it is possible to write theequation of deformation motion as:

(Eqn. 112)

Integration of this equation leads to the following time-dependent particle distortionequation:

(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

%Pr2----------- y 8,%Pr3----------- y+

2%g V slip2

3%g r2---------------------= =

y t( ) W eC e t tD⁄– y0 W eC–( ) -tcosy0------

y0 W eC–-tD

-----------------------+2 34 5 -tsin++=

tD2%Pr2

CdµP--------------=

-2 Ck,

%Pr3----------- 1td

2----–=

W eC WeCf

CkCb-------------=

y0 y0

y0 y0



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 thedeviation of the particle equator from its equilibrium position has become larger than halfthe droplet radius.

The Sauter mean radius of the child droplets after breakup is calculated from the followingexpression:

(Eqn. 117)

that is based on the conservation of surface energy and energy bound in the distortion andoscillation 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 afterbreakup. 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

%PrP,Parent3

,------------------------ y02 6K 5–

120----------------2 34 5+ +=

V N CvCbr y=

\

\2---tan

V NV slip-----------=

y 0( ) y 0( ) 0= =



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 standardTAB 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

k1- W e W eth

k2- We W e W et>;=?

=



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 thatis 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 sprayparameters such as the penetration depth, as well as of local parameters such as thecross-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⁄+ -2

y2------=

-2 Ck,%PrP Parent,---------------------------=

CD

y 0( ) y 0( )

y 0( )

y 0( ) 1 W eC 1 -tbucos–( )–[ ] --tbusin-------------------=



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 ofthe jet breakup.

In addition to the TAB model constants, the following user accessible model constants areavailable 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 andBreakup Model, CAB. Identical to the ETAB model, the following equation is used todetermine 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 areavailable 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 C%P%F------

dP 0,V P 0,-----------=

y 0( )

K1

K2

Wet

rP Child,rP Parent,-------------------- e

K brt–=

Kbr

Kbr

k1- 5 W< e 80<

k2- We 80 W< e 350<

k3-W e3 4⁄ 350 We<;]]=]]?

=



Table 4 CAB breakup model constants and their default values

Dynamic Drag Models

Many particle drag models assume that the droplet remains spherical throughout thedomain. However, this is not always the case and the particle shape may be distortedsignificantly. In the extreme case, the particle shape will approach that of a disk. The dragcoefficient is highly dependent on the particle shape and it is therefore desirable to modifythe standard drag laws to account for the effects of droplet distortion.

In ANSYS CFX, the following models are implemented that modify the drag coefficientdepending 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 theTAB, ETAB and CAB breakup models.

Hsiang [109] Based on experimental findings, Hsiang and Faeth suggest to use the following relation todescribe the drag coefficient of the deformed droplet within a Reynolds number range of1000 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 theparticle 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=



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 dragcoefficient 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 draglaw 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 64GRe----------+=

f E2 1–=



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 isexceeded for the first time.

Figure 5 Penetration depth

\

SP

SN

SR



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

!



Combustion Theory

Introduction


• Transport Equations (p. 228)

• Chemical Reaction Rate (p. 228)

• Fluid Time Scale for Extinction 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 ANSYSCFX-Solver.

This chapter extends the ideas covered in Multicomponent Flow. You should be familiarwith the multicomponent flow chapter before reading this section.

The following sections outline the basis of the implementation of combustion modeling inANSYS CFX. First, the transport equations for energy and the components are revisited, thenthe chemical reaction rate computation is described, and finally, the computation of the

Combustion Theory: Transport Equations


rate of progress of a chemical reaction is explained in the context of the Eddy Dissipationand Finite Rate Chemistry Models. Extinction is modeled by setting the reaction rate locallyto zero.

Transport Equations

Combustion models in ANSYS CFX use the same algorithm used for Multicomponent Fluidwith the addition of a source/sink term due to chemical reactions. The equation of transportfor 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 involvingNC 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.

6 %Y I( )6t-----------------

6 %u jY I( )6x j

-----------------------+ 66x j-------- $Ieff

6Y I6x j---------2 3

4 5 SI+=

EkIW I

I A B C …, , ,=

N C

V EkIWW I

I A B C …, , ,=

N C

Vk

EkI I k

SI I

I

SI WI EkIWW EkI

W–( )Rkk 1=

K

V=

Rk k

Combustion Theory: Fluid Time Scale for Extinction Model


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 tothe transport processes in the flow. When reactants mix at the molecular level, theyinstantaneously form products. The model assumes that the reaction rate may be relateddirectly to the time required to mix reactants at the molecular level. In turbulent flows, thismixing 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 problemswhere reaction rates are fast compared to reactant mixing rates.

In the Eddy Dissipation model, the rate of progress of elementary reaction k, is determinedby 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 reactantcomponents.

.t .c<

.c

.Kolmogorovv"--=

.mixingk"--=

"

rate "k--a

Rk A"k--min I[ ]

vkIW--------

2 38 94 5

=

Combustion Theory: The Finite Rate Chemistry Model


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. Forboth single step and multi-step reaction schemes, it is turned off by default, (set to -1), butmay be turned on by explicitly setting the model coefficient B to a positive value (althoughthis 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 limitis approached. This is implemented by an additional bound added to the minimumcondition 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 ofprogress 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 "k--

I[ ]WIPV

/kIWW WI

PV--------------------------

2 38 98 98 94 5

=

Rk MFT, A"k--CMFT=

CMFT max Tmax T–( ) 0 K[ ]{ , }%CP

HRB-----------:=

CMFT

CP

HRB

Rk Fk I[ ]r W

kI

I A B …, ,=

N C

l Bk I[ ]r WW

kI

I A B …, ,=

N C

l–2 38 94 5

=

Combustion Theory: The Finite Rate Chemistry Model


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 reactionorder is equal to the stoichiometric coefficient for elementary reactions, but it can bedifferent for certain global reactions.

The only built-in formula for the forward and backward rate constants assumes an Arrheniustemperature 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 describedearlier 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 forindividual species by setting Option to Efficiency Factor List and listing thecomponents 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

T'k Ek

RT-------–2 34 5exp=

Bk

Ak

T'k Ek

RT-------–2 34 5exp=

'*

Rk

Ak 'k Ek

&ki I i[ ]i 1=

N

V

&ki

&ki

&ki 0=

Combustion Theory: The Combined Eddy Dissipation/Finite Rate Chemistry Model


The Combined Eddy Dissipation/Finite Rate Chemistry Model

The effective reaction rate, for the combined model, is computed to be the minimum of theFinite Chemistry Rate and the Eddy Dissipation rate.

The Theory documentation for this model is the same as for others:



Combustion Source Term Linearization

The combustion source terms may have a dominant influence on the solution of the scalarand energy equations. Thus, it is important to treat the combustion source terms carefullyin order to obtain robust convergence of the fluid flow.

A property of multicomponent fluids is that the mass fraction of any given component isbounded between 0 and 1. Combustion tends to drive reactant concentrations towards thelower limit and product concentrations toward the upper limit. If the timestep is large, thecombustion sources may cause scalars to exceed these bounds. Thus, the sources may needto be moderated to maintain physically realistic mass fractions.

The combustion sources in ANSYS CFX have been linearized to prevent the formation ofnegative 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 Ifrom 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 `--------------------2 34 5 I

RI RI+2 1 I–( )**------------------------2 34 5 1 I–( )+=

I ` max ( I,( )= 1 I–( )** max ( 1, I–( )=

"

source RI if I 1 (–i=

source RI1 I–( )(----------------=

products 1b source 0b

Combustion Theory: The Flamelet Model


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 asteady state solution than would be possible without the linearization.

The Flamelet Model

The Flamelet concept [37] for non premixed combustion describes the interaction ofchemistry with turbulence in the limit of fast reactions (large Damköhler number). Thecombustion is assumed to occur in thin sheets with inner structure called Flamelets. Theturbulent flame itself is treated as an ensemble of laminar Flamelets which are embeddedinto the flow field.

The Flamelet model is a non equilibrium version of the classical “Burke-Schumann” limit. Itadds new details to the simulation of combustion processes compared to other commoncombustion models for the price of the solution of only two scalar equations in the case ofturbulent flow. An arbitrary number of intermediates may be specified as long as theirlaminar chemistry is known.

The main advantage of the Flamelet model is that even though detailed information ofmolecular transport processes and elementary kinetic reactions are included, the numericalresolution of small length and time scales is not necessary. This avoids the well-knownproblems of solving highly nonlinear kinetics in fluctuating flow fields and makes themethod very robust. Only two scalar equations have to be solved independent of thenumber of chemical species involved in the simulation. Information of laminar model flamesare 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 differentLewis numbers of the chemical species.

The coupling of laminar chemistry with the fluctuating turbulent flow field is done by astatistical method. The PDF used can in principle be calculated at every point in the flowfield by solving a PDF transport equation as shown by Pope and many others. The mostoften mentioned advantage of this method is that the non-linear chemical source termneeds no modeling. Even though the method avoids some modeling which is necessary ifusing moment closure, it still requires modeling of some of the most important terms, inparticular, the fluctuating pressure gradient term and the molecular diffusion term. If

source RI if I (i=

sourceRI I( )(--------------=

reactant 0b source 0b



combustion occurs in thin layers as assumed here, the molecular diffusion term is closelycoupled to the reaction term and the problem of modeling the chemical source term is thenshifted towards modeling the diffusion term.

However, there is no source term in the mixture fraction equation, which is the principaltransport equation in the Flamelet model. Therefore, a presumed beta-PDF, which is acommonly accepted choice, is used here. Additionally, this avoids the extremely largecomputational 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 modelcan be combined with a model for reaction progress. For details, see Burning VelocityModel (Premixed or Partially Premixed) (p. 239).

Fluid properties, including temperature and density, are computed from the meancomposition of the fluid in the same way as for other combustion models, such as the EddyDissipation model.

The Flamelet model as implemented in ANSYS CFX can be applied for non-adiabaticconfigurations. The only limitation is that changes in the composition of the fluid due todifferent temperature and pressure levels are not accounted for. However, the effect of heatloss and pressure on density and temperature is taken into account. For heat lossesoccurring in many combustion devices, the influence of on composition is sufficiently smallto be neglected.

In a large number of industrial combustion devices, pure non-premixed combustion is lesspresent than premixed or partly premixed combustion. In ANSYS CFX, a model for premixedand partially premixed combustion is available, which involves the Flamelet model as asub-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. Whileconvective and diffusive time scales are of the same order of magnitude, the chemical timescales are much smaller for typical combustion processes of interest. Several approaches totreat chemical reactions have been developed and tested during the last decades.

The assumption of local chemical equilibrium has often been used in modeling the fastchemistry regime. For hydrocarbon flames, however, the assumption of local chemicalequilibrium 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 nonequilibrium effects are flame extinction, lift-off and blow-out.

Lei 1=



Another well known approach is the flame sheet model of Burke and Schuman oftencharacterized as `mixed is burned'. Here, only the mixture of the reactants is calculated andthe 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. Againnon equilibrium effects are not taken into account, which are important if the stronglyvarying 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. Heanalyzed the inner structure of the thin laminar flame sheet, referred to here as a Flameletusing an asymptotic description with a large Damköhler number as the expansionparameter. 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 thinreaction sheet with well defined structure is called “Flamelet” from now on. A more simpledescription of flamelets is possible by using the mixture fraction, which is the sum of allelementary 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 conservationequation for the mixture fraction Z can be derived by summing all species conservationequations, and the chemical source terms therefore cancel exactly. The mixture fraction isnot 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

V=

Y i Z j j

M aij j i

ZZFuel

ZFuel 1,---------------- 1

ZOxidiserZOxidiser 2,-----------------------–= =

6 %Z( )6t--------------- 6 %Z( )

6x j---------------+ 6

6x j-------- %D 6Z

6x j--------2 3

4 5=



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 isintroduced here. One of its coordinates is locally perpendicular to the surface ofstoichiometric mixture.

The transformation is shown for the temperature equation as an example. The 3 terms at theright hand side represent chemical reactions, radiation and the transient pressure gradient,respectively. The last is important for combustion involving fast changing pressure such asin 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=

%6T6t------- %/&

6T6x&--------- 6

6x&--------- %D 6T

6x&---------2 3

4 5–+Qkcp-------k

qRcp----- 1

cp-----6P

6t------+ +k 1=

r

V=

66t-----

66.-----

6Z6t-------

66Z-------+=

66xk-------- 6

6Zk--------- 6Z

6xk-------- 66Z------- k 2 3,=( )+=

66x1-------- 6Z

6x1-------- 66Z-------=

Le +%cpD-------------=

% 6T6.------- u2

6T6Z2--------- u3

6T6Z3---------+ +2 3

4 5 6 %D( )6x2---------------- 6T

6Z2---------– 6 %D( )

6x3---------------- 6T

6Z3---------–

%D 6Z6x&---------2 34 5 262T

6Z2--------- 2 6Z6x2-------- 62T

6Z6Z2---------------- 2 6Z

6x3-------- 62T

6Z6Z3---------------- 62T

6Z22--------- 62T

6Z32---------+ + + +–

1%cp-------- himi

qRcp----- 1

cp-----6P

6t------+ +i 1=

n

V=



Since the Flamelet is assumed to be thin, only gradients normal to the surface ofstoichiometric mixture are large, and all terms without a second derivative in respect to themixture fraction Z can be neglected. When this is done formally by introducing a stretchedcoordinate, it turns out that the remaining equation is essentially one dimensional. Thesame 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 sothin that the production of heat cannot balance the heat loss caused by conduction. Thetemperature drops to unburnt values and the reactions freeze. `Freeze' means they are atlower temperatures and are much slower than the fluid time scales.

The important conclusion of this derivation is that flamelet structures in the presence of fastchemistry can be described by one dimensional model flames. This will be used to modelreacting turbulent flow fields.

Coupling of Laminar Flamelet with the Turbulent Flow Field

In the turbulent flow field, the Favre averaged (tilde superscript) mixture fraction equationis 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)

6T6t-------

Zst2------62T6Z2---------– 1

%cp-------- himi

˙ qRcp----- 1

cp-----6P

6t------+ +i 1=

n

V=

Zst

Zst 2Dst CZ( )st2=

Z Zq=

6 %Z( )6t---------------

6 %u jZ( )6x j

--------------------+ 66x j-------- µ

µt,Z------+2 3

4 5 6Z6x j--------

; <= >? @

=

6 %Z''2˜

( )6t--------------------

6 %u j˜ Z''2˜

( )6x j

-------------------------+ 66x j-------- µ

µt,

Z''2---------+

2 38 94 5 6Z''2

˜

6x j-----------

; <] ]= >] ]? @

2µt,Z------ 6Z

6x j--------2 34 5 2

%Z–+=



The structure of this equation is similar to the mixture fraction equation itself except for lasttwo 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, seeFlamelet Libraries (p. 239). The CFD solver looks up the preintegrated values from thelibrary.

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 librariescreated with CFX-RIF. For details, see CFX-RIF (p. 275 in "ANSYS CFX-Solver ModelingGuide").

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 mixturefraction. This is exact only for stoichiometric mixture or for vanishing variance of mixturefraction (perfectly premixed case). However, many radicals of interest, for example, OHradicals, have significant concentrations only at stoichiometry and in its surrounding, which

Z

Z CZ"k--Z''2=

,Z 0.9= ,Z''2

0.9= CZ 2.0=

Y i˜ Y i

˜ Z Z''2 Zst,˜

,( ) Y i Z Zst,( ) PZ Z''2

˜,

Z( )6Z:0

1

7= =

P

P '

PZ Z''2

˜,

Z( ) Za 1– 1 Z–( )b 1–

Ua 1– 1 U–( )b 1– Ud0

17-----------------------------------------------------=

a Z Z 1 Z–( )

Z''2˜--------------------- 1–

2 38 94 5

b, 1 Z–( ) Z 1 Z–( )

Z''2˜--------------------- 1–

2 38 94 5

= =

Z Zst

Combustion Theory: Burning Velocity Model (Premixed or Partially Premixed)


makes this approximation acceptable. In principle, could be derived from the solution

fields, but this would introduce errors, too, because it would require either additionalmodeling or averaging over the computational domain.

Flamelet Libraries

A flamelet library provides the mean species mass fractions as functions of mean mixturefraction, variance of mixture fraction and scalar dissipation rate:

(Eqn. 38)

A separate Flamelet library is required for each fuel and each combination of fuel/oxidiserinlet temperatures and pressure level. The libraries shipped with ANSYS CFX are listed inTable 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 twoindependent 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: LaminarFlamelet 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)

Zst

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 Zst,˜

,( )=

TFuel TOxid

Y i fresh,

Y i burned,

c

Fuel Oxidiser Productsb+



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 isjustified if the chemical reaction is fast compared to the integral turbulent time scales of theflow.

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 computedaccording 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 becauseburnt 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=

6 % c( )6t--------------

6 %u j˜ c( )

6x j-------------------+ 6

6x j-------- %D

µt,c-----+2 3

4 5 6 c6xj------- -c+=

,c

,c 0.9=

Y i,burned Z 0=( ) Y i,fresh Z 0=( )=

Y i,burned Z 1=( ) Y i,fresh Z 1=( )=

cc

c 0=

c 1=

c 0= c 1=c



• 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 flowsplit, 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 tosolving for reaction progress directly. In other words, introducing the weighted reactionprogress overcomes the boundary value issue for reaction progress without changing themodel. Reaction progress can be restored from mixture fraction and the weighted reactionprogress 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 tozero:

c 0=

c 1=

c 1= c 0=

c 0= Z 0= c 1= Z Zst=

c &= Z & Zst:=

& 1<

c 1= Z 0=

F

F Z 1 c–( ):=

Z 0=

F 0= Z c

6 %F( )6t---------------

6 %uj˜ F( )

6xj--------------------+ 6

6xj------- %D

µt,F------+2 3

4 5 6F6xj------- 2 %D

µt,F------+2 3

4 5 6Z6xj------- 6 c

6xj-------:2 3

4 5 Z-c–+=

,F 0.9=

c Z F–( ) Z⁄=

c Z 0=

Combustion Theory: Burning Velocity Model (BVM)


as

Default for ANSYS CFX, Release 11.0 is to solve for weighted reaction progress. Expertcontrol (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 toclose 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 ofspecies 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 twosignificant 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 asflame position. Thus, it is easier to derive and fine-tune accurate models for the burningvelocity model than to do so for approaches based on molecular reaction rates.

c 0b 6 c6xj------- 0b, Z 0b

-c Sc66xj------- %D( ) 6 c

6xj-------2 3

4 5–=

Sc %uST C c=

%u

Sc

sT

sT

sT

Combustion Theory: Burning Velocity Model (BVM)


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.

• ValueThis option allows the user to specify directly the stoichiometric mixture fraction usedfor calculating the equivalence ratio.

• ReactantsFor 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 libraryrequiring no additional information. The numerical procedure is described below.

0

0 1=

0 1> 0 1<0 0= 0 F=

Zst

Z

0 Z1 Z–------------

1 Zst–Zst

---------------:=

Zst

EFFuel EOxO2+ (products)b

E&X&&V (products)b

Combustion Theory: Laminar Burning Velocity


Figure 1 shows the qualitative behavior of mass fractions for fuel and oxygen plotted overmixture 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 abovethe stoichiometric mixture fraction: linear decay on the lean side and constantly zero on thefuel side. Obviously, the curvature of the curve is close to zero except near the sharp bendat stoichiometric mixture fraction. This observation is generalized to establish the followingprocedure

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 tocheck for plausibility of the calculated value, which is reported to the solver output file. Fordetails, see ANSYS CFX Output File (Combustion Runs) (p. 50 in "ANSYS CFX-Solver ManagerUser'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. Theburning velocity relative to the burnt fluid will be higher by a factor equal to the expansion

ratio, .

Zc

Zst Zcd62Y O2

Zc( )

6Z2-------------------------, max0 Z 1h h62Y O2

Z( )

6Z2-----------------------=

sL

sLburnt sL %u %b⁄:=



Physically, the laminar burning velocity depends on the fuel, the equivalence ratio, thetemperature of the unburnt mixture (preheating) and on pressure. Depending on theconfiguration in the simulation, it may be possible sometimes to neglect preheat andpressure dependencies. However, for partially premixed combustion, it is very important toaccount for the dependency on equivalence ratio. Specifically, the flammability limits haveto 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 mixturefraction. Properties of the unburnt mixture (temperature, density, specific heat capacity,thermal conductivity) may be used in the expression in order to account for preheating ormixture 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 quadraticpolynomials in equivalence ratio:

(Eqn. 54)

(Eqn. 55)

When all three coefficients are set to zero, then the preheat dependency or the pressuredependency 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---------2 34 5 & p

pref--------2 34 5 ': :=

& a0 a10 a202+ +=

' b0 b10 b202+ +=

0flam,l 0flam,r



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 withquadratic decay described above. Predefined sets of coefficients are provided for severalhydrocarbon fuels. The fuel type is characterized by the number of carbon atoms in the fuelmolecule, 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 Burning

Velocity Correlation

For fuels with other carbon indices the coefficients are obtained by linear interpolation ofthe provided values. Fit range, preheat dependency and pressure dependency are modeledindependent of the fuel (Table 3).

sL0 s00 s10 s20

2 s303 s40

4 s505+ + + + +=

0fit,l 0 0fit,rh h

smax0

0max

Cdecay

sL0 s0

max Cdecay 0 0max–( )2–=

0fit,l 0 0fit,rh h

s0max

0max

sL0 s0

max 0 0flam,l–0max 0flam,l–------------------------------2 34 5 a 0flam,r 0–

0flam,r 0max–-------------------------------2 34 5 b

: :=

a 20max 0flam,l–0flam,r 0flam,l–---------------------------------:= b 2

0flam,r 0max–0flam,r 0flam,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 0max 0flam,l 0flam,r

Combustion Theory: Turbulent Burning Velocity


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 highturbulence, the opposite effect may occur, leading to a decrease in the effective burningvelocity because of local extinction. A model is required to describe the turbulent burningvelocity 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 burningvelocity:

(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

0fit,l 0fit,r a0 a1 a2 b0 b1 b2

sT

sL

sTburnt sT %u %b⁄:=

sT AGuW3 4⁄ sL1 2⁄ +u

1 4⁄– lt1 4⁄=

A A 0.5=A 0.6=

G

" "cr

" "cr> " "cr<

"

G 12--erfc 1

2,----------- "cr "⁄( ) ,

2---+ln2 34 5–=



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 laminarcases because the smallest turbulent eddies, which are responsible for the largest strainrates, do not persist long enough to quench a flame front locally. Furthermore, differentmodel problems may result in significant variation of the critical values obtained. For thesereasons, 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 notedthat these recommended values are for atmospheric temperature and pressure.

erfc , µstr lt T⁄( )ln=

" µstr

µstr 0.28=

+u

uW 23--k=

lt k3 2⁄ "⁄=

T E3 4⁄ "1 4⁄⁄=

"cr gcr

E

"cr 15Egcr2=

gcr

gcr

.ch

gcr 6000 s 1–[ ]= 10000 s 1–[ ]



Table 4 Default Model Coefficients for the Zimont Turbulent Burning VelocityCorrelation

Peters Correlation

The Peters correlation for turbulent burning velocity was originally developed for theG-equation combustion model (N. Peters [37]). For ANSYS CFX, it was adopted to be usedwith 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 theunburnt mixture indicated by the subscript u. The default values for the model constants arelisted in Table 5.Table 5 Default Model Coefficients for Peters Turbulent Burning Velocity

Correlation

Parameter Default Value0.5

0.28

10000

Aµstr

gcr s 1–

Coefficient Default Value0.37

0.78

2.0

1.0

sT sL 1 ,t+( )= ,t A llF----– A l

lF----2 3

4 5 2 u'lsLlF---------++= u' 2

3--k= Aa4b3

2

2b1----------=

B a4b32=

l lF

l a1u'3

"------=

lFDsL----

+ cp⁄( )u%usL

-------------------= =

a1

a4

b1

b3

Combustion Theory: Spark Ignition Model


Mueller Correlation

The correlation by Mueller et al. follows an approach similar to the Peters correlation. Itaccounts for variation of the average laminar burning velocity due to turbulent fluctuationsof 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 withthe Turbulent Flow Field (p. 237). Table 6 lists the default values for the model coefficients.Table 6 Default Model Coefficients for Mueller Turbulent Burning Velocity

Correlation

Spark Ignition Model

The purpose of the spark ignition is two-fold: First, it is required in order to provide theappropriate 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 Value1.5

0.8

1.0 [1/s]

1.0

sT sL 1 ,t+( ) 1 & ZZq-----–2 3

4 5: := ,t b1u'

sL,st-------- b2

u'sL,st--------+= u' 2

3--k=

sL,st sL Zst( )=

0 1=

sL P Z( ) sL Z( ):( ) Zd0

17=

b1

b4

Zq

&

rK

t ignite

V initial

tdd rK t( )

%u%b----- sT,k:=

Combustion Theory: Phasic Combustion


where

The spark kernel radius is mapped onto the three-dimensional flow field by averaging thereaction 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 modificationaccounting 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 eachindividual phase. The combustion models are solved in exactly the same way as for a singlephase simulation.



• 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

4G------------------3=

rtrans xspark

crK

rtrans-----------2 34 5 3

= x xspark– rtransh

c 0= x xspark– rtrans>

rtrans

sT,k max sL sT2rk----Dt–,2 3

4 5=

DtCu,t------ k2

"-----:=

sL sT k "

%b %u

Combustion Theory: NO Formation Model


NO Formation Model

The NO formation model is fully integrated into the ANSYS CFX reaction and combustionmodule. 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 fromthe extensions implemented for the NO model (Arrhenius with Temperature PDF reactionrates, variable pre-exponential factor for Arrhenius rates).

Formation Mechanisms

The formation of NOx is a complicated process involving several different mechanismswhich 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 reburnare predefined in the REACTIONS database. It is possible to add reactions for othermechanisms, or modify the provided mechanisms, using the Reaction details view inANSYS 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 temperaturesabove 1800 K. The NO is formed from the combination of free radical O and N species, whichare in abundance at high temperatures. The two-step mechanism, referred to as theZeldovich 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 Zeldovichmechanism.

O N2 NO N+b+

N O2 NO O+b+

OH N NO H+b+



The rates of each of these three reactions (using the subscripts 1,2,3 to describe the threereactions) 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 Nradical is assumed to be oxidized by reaction (Eqn. 70) in the Zeldovich mechanism and alsoby reaction (Eqn. 71) in the extended Zeldovich mechanism. Either way, these secondoxidation 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 betaken 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 Chemistrymodel (FRC), O radical concentrations usually are not known directly but must be derivedfrom other quantities. Here, the O radical concentration is estimated from the molecularoxygen dissociation,

(Eqn. 77)

(Westenberg, 1975):

(Eqn. 78)

k1 1.8 1011:( ) 38370T--------------–2 3

4 5exp=

k2 6.4 109:( ) 3162T-----------–2 3

4 5exp=

k3 3.0 1013:=

kmol m3⁄ s⁄[ ]

kg m3⁄ s⁄ SNO thermal,

SNO thermal, WNOkthermal O[ ] N2[ ]=

kthermal 2k1=

WNO

12--O2 Ok

O[ ] 12567 kmol1 2⁄ m 3 2⁄– K 1 2⁄[ ] T 1 2⁄– exp(-31096 K T )⁄ O2[ ]1 2⁄:=



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 concentrationthat is too high to be explained with the Zeldovich mechanisms. Hydrocarbon radicals canreact with molecular nitrogen to form HCN, which may be oxidized to NO under lean flameconditions.

(Eqn. 81)

(Eqn. 82)

The complete mechanism is very complicated. However, De Soete (see also Peters andWeber, 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 thefollowing 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. HCNis 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+b+

HCN O2 NO ...+b+

SNO prompt,

SNO prompt, WNOkprompt O2[ ]1 2⁄ N2[ ] Fuel[ ] W%------2 3

4 5 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[ ]=



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 oxidizeall of the fuel, the excess fuel may lead to reduction of NO. This process can be described bya global reaction:

(Eqn. 90)

One should keep in mind that this is only a global representation. The physical processactually occurring is much more complicated. The real process involves many intermediatecomponents appearing during combustion of the fuel, e.g., CHx radicals, which attack theNO.

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, theglobal NO reburn reaction is:

(Eqn. 91)

HCN O2+ HCO NO+b

HCN NO+ HCO N2+b

HCO 34--O2+ CO2

12--H2O+b

R1 3.5 1010 s 1–[ ] HCN[ ] XO2( )& 67kcal

mol----------2 34 5 RT( )⁄–2 3

4 5exp: : : :=

& 11 XO2

0.007⁄( )2+------------------------------------------=

R2 3 1012 s 1–[ ] W%------ HCN[ ] NO[ ] 60kcal

mol----------2 34 5 RT( )⁄–2 3

4 5exp: : : :=

R3 8.3 106 s 1–[ ] HCO[ ] 0.3– O2[ ]1.3 30kcalmol----------2 3

4 5 RT( )⁄–2 34 5exp: : : :=

XO2O2 W

NO EFFuel+ 12--N2 ECO2

CO2 EH2OH2O+ +b

NO 14--CH4+ 1

2--N214--CO2

12--H2O+ +b



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 turbulencesystems, fluctuations can have a dominant impact on the NO formation rate. For both thethermal and prompt NO mechanisms, there is a strong dependence of the rates on thetemperature due to their high activation energy. Thus, temperature fluctuations,particularly positive fluctuations, can dramatically increase the NO formed in flames. Thesetemperature fluctuations are included in ANSYS CFX using a statistical approach.

In order to determine the mean rate for NO formation, a presumed probability densityfunction (presumed PDF) method is used to compute the weighted average of the reactionrate:

(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; 2300K], 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–[ ] W%------ NO[ ] Fuel[ ] 9460 K[ ]–( ) T⁄( )exp: : : :=

k 1Tu Tl–----------------- k T( ) P T( )• Td

Tl

Tu

7=

Tl Tu;[ ]

P T

T''2˜

P'

P T( ) ca 1–( ) 1 c–( )b 1–

Ua 1– 1 U–( )b 1– Ud017-----------------------------------------------------=

a c c 1 c–( )g--------------------- 1–2 3

4 5 b, 1 c–( ) c 1 c–( )g--------------------- 1–2 3

4 5= =

cT Tl–Tu Tl–----------------- c,

T Tl–Tu Tl–----------------- g, T''˜

Tu Tl–( )2-------------------------= = =



For vanishing temperature variance (vanishing temperature fluctuations), the beta functionis approaching to a single Dirac peak (delta-function). In the limit, the integrated reactionrate is the same as for that for the standard Arrhenius rate. For very large fluctuation, thebeta function goes towards a double Dirac peak, and for small to medium temperaturevariance 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 NOformation 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 issolved:

(Eqn. 97)

Default values for the model co-efficients are and . .

The above equation is missing some physical aspects namely the production oftemperature fluctuations due to heat release by chemical reaction. Heat release isfluctuating, too, because of turbulent fluctuations of the reactants. However, in the currentmodel temperature variance is only needed as input to another model: The construction ofa probability density function with presumed shape. For this purpose, the above equationprovides sufficient accuracy.

For convenience of setting up a case, the temperature variance equation can be run withoutspecifying BC or IC data. In the first release, these are not offered by the GUI (ANSYSCFX-Pre). If absent, zero fluctuations are assumed at inlets, openings and walls withspecified temperature. At walls with specified heat flux or transfer coefficient, the default BCfor temperature variance is zero flux.

Model Control Since the model for NO formation is implemented by means of REACTION objects, you havefull control of all aspects of the model. ANSYS CFX provides the same flexibility for the NOmodel 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 formationreactions are defined by means of REACTION objects, they may be edited either in ANSYSCFX-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 integrationrange for the presumed PDF. It is specified by the Lower Temperature and Upper

Temperature parameters. These appear in the FORWARD REACTION RATE object whenOption is set to Arrhenius with Temperature PDF. The predefined reaction schemescome with the range set to the interval [300 K, 2300 K]. It is recommended to adjust this tothe maximum temperature range occurring. For many systems, this temperature range isdefined by the minimum inlet temperature and the adiabatic flame temperature.

T''2˜

6%T''2˜

6t---------------6%u j˜ T ''2˜

6x j--------------------+ 6

6x j-------- µ

µtPrt-------+

2 38 94 5 6T ''2

˜

6x j-----------

; <= >? @

CprodµtPrt------- 6T

6x j--------2 38 94 5

2

Cdiss%"k--T ''2˜

–+=

Cprod 2.0= Cdiss 2.0=

Combustion Theory: Chemistry Post-Processing


User Defined NO Formation MechanismsThe NO formation model can be extended to a user's specific needs by adding appropriatereactions 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 oftemperature.

It is also possible to select each NO formation path individually (thermal, prompt, fuelnitrogen, reburn, or possibly user defined). Simply add only those reactions to the mixturematerial that you want to account for.

Chemistry Post-Processing

The components and reactions that are post-processed are one-way coupled to the mainsimulation. This means that there is no effect on mixture properties or heat release. The listof 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 reactionturnover are small relative to the bulk mixture. A typical application is the simulation ofpollutants, e.g., NO formation. The transport equations for post-processing components aresolved at the end of the simulation after the solving of the main equations. When thepost-processing reactions require a temperature variance equation (reaction rate option isArrhenius with Temperature PDF) and the variance equation was not solved during themain simulation, then temperature variance will be post-processed as well. In this case, thetemperature variance equation will be solved first to convergence or maximum number ofiterations and is then followed by the post-processing components group.

Combustion Theory: Soot Model


Soot Model

In the Magnussen soot model (Magnussen and Hjertager [46], it is assumed that soot isformed from a gaseous fuel in two stages, where the first stage represents formation ofradical 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 onmean 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 mustbe assigned to one of the fluids.

Soot Formation

Formation of nuclei and soot particles is computed following the empirical models of Tesneret al. [45]. The source terms are formulated in terms of particle number concentrations fornuclei:

(Eqn. 100)

and soot particles:

(Eqn. 101)

Where [part/mol] is Avogadro’s number and:

(Eqn. 102)

XN

Y s˜

6 %XN˜( )

6t-------------------6 %u j

˜ XN˜( )

6x j-------------------------+ µ

µtPrt-------+2 3

4 5 6XN˜

6x j-----------

; <= >? @

Snuclei f, Snuclei c,+ +=

6 %Y S˜( )

6t------------------6 %u j

˜ Y S˜( )

6x j-----------------------+ µ

µtPrt-------+2 3

4 5 6Y S˜

6x j---------

; <= >? @

Ssoot f, Ssoot c,+ +=

CN % A XN [part/m3]: :=

CS %Y Smp------- [part/m3]:=

A 6.02214199 1023:=

mp %sootGd3 6⁄=



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 physicaldimensions. 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)

%soot d

Snuclei f,

Ssoot f,

Snuclei f, n0 f g–( )CN g0CN CS–+=

Ssoot f, mp a bCS–( )CN=

n0

n0 a0 f c%Y fuel T A 0, T⁄–( )exp=

f c f

g g0

a b

%soot 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



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 combustionthat were made for fuel consumption. For example, when the Eddy Dissipation model isapplied for the fuel consumption reaction, combustion of nuclei and soot particles isassumed to operate at the fast chemistry limit. No special model for turbulence effects isrequired for the nuclei and soot combustion rates as this is already accounted for in thecomputation of the fuel reaction rate.

Turbulence Effects

The above reaction rates are applicable to laminar flow. In turbulent systems, fluctuationscan have a significant impact on nuclei and soot formation because of the non-linearity ofthe respective source terms. To account for the effect of turbulence on soot formation, theEddy Dissipation Concept (EDC) developed by Magnussen [44] is applied.

The Eddy Dissipation Concept (EDC) is a reactor concept that identifies a reactor, where thecombustion of fuel takes place, related to the fine structures in turbulence. This reactor istreated 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 theturbulence quantities.

Local balance equations are solved to compute the temperature and the concentrations ofnuclei and soot particles in the fine structures and in the surrounding fluid, respectively. Themean source terms are then computed assuming a bimodal distribution of the fluidbetween the fine structures and the surrounding. For a detailed explanation of the EddyDissipation Concept, see (Magnussen, 1989).

The EDC procedure need not be applied to the soot combustion terms, because turbulenceis already accounted for in the computation of the fuel reaction rate. For details, see SootCombustion (p. 261).

Snuclei c, Ssoot c,

Sfuel

Snuclei c, SfuelXN˜

Y fuel˜-----------=

Ssoot c, SfuelY S˜

Y fuel˜-----------=





Radiation Theory

Introduction


• 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 thealgorithms 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 thesource term, S, for the energy equation, and the radiative heat flux at walls, among othersquantities of interest. You should restrict yourself to coherent time-independent radiationprocesses. This is normally a very good approximation of situations likely to be met inindustrial applications because the time scale for radiation to come into local equilibrium isvery short and the temperatures are relatively low.

The spectral radiative transfer equation (RTE) can be written as:

(Eqn. 1)

dIE r s,( )ds---------------------

KaE KsE+( )IE r s,( )– KaEIb E T,( )KsE4G-------- dIE r s',( )# s s'•( ) ^' S+d

4G7+ +

2 38 94 5

=

Radiation Theory: Radiation Transport


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, andfrequency, the formal solution of the radiative transfer equation is very time consuming andusually accomplished by the use of approximate models for the directional and spectral

Ib

IE

^

#

IE rw s,( ) "E rw( )Ib E T,( )%w rw( )

G----------------- IE rw s',( ) n s'• ^'dn s' 0<•7+=

"E

IE rw s,( )

"E rw( )Ib E T,( )%E

d rw( )G---------------- IE rw s',( ) n s'• ^'d

n s' 0<•7 %E

s rw( )IE rw ss,( )++2 38 94 5

=

%Ed 1 "E–( )

%Es 1 "E–( )

%E %Ed %+ E

s1 "E–( )

ss



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 temperatureand because of this strong dependence, radiation is usually unimportant or totallydominant for heat transfer.

The sun, for example, is approximately a blackbody at a temperature of 5700 K. Thespectrum peaks in the yellow part of the visible spectrum.

Eb E T,( ) 2Gv2

c2----------- n2hv

ehv kBT⁄( )

1–------------------------------ W m 2– Hz 1–[ ] GIb E T,( )= =

n

h

c

kB

x hv kBT⁄=

Eb x T,( ) n2,T4 hkT-------2 34 5 15

G4-----x3

ex 1–-------------

2 38 94 5

=

,

, 2G5k4

15h3c2----------------m

Eb

Eb T( ) Eb x T,( ) xd0

F

7 n2,T 4= =



Combustion temperatures are typically 1000 - 2000 K, with spectrum peaks in the nearinfra-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 spectralradiative 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 thermodynamicequilibrium.

OpticalThickness

Optical thickness is a measure of the ability of a given path length of gas to attenuateradiation 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 4kBTn

qv

qER r n,( ) s n•( )IE r s,( ) ^sd7=

qERC•–( ) Ka GE 4EbE–( )=

GE IE ^d s7m

qRC• qER Ed

0

F7 K avGv vd

0

F7 4 K avEbE vd

0

F7–= =

K a 0= qRC• 0=

*+ S( ) K+ S*( ) S*d0

S7=

*+ S( ) S

K+ S



Radiation Through Domain Interfaces

If radiation is included through conducting solids, then usually the difference in refractiveindices between the fluid and solid determines the amount of reflection and refraction thatoccurs. The probability of being reflected is given by Fresnels’ equation

(Eqn. 14)

The fraction of the electromagnetic wave that is reflected normally depends on thepolarization 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 averageresult. The two extreme polarizations are termed transverse electric (TE) and transversemagnetic (TM), and describe the orientations of the electric and magnetic vectors relative tothe 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 isproportional to the square of the wave amplitude. Thus, the average reflection coefficient isgiven as:

(Eqn. 17)

and the probability of being transmitted is:

(Eqn. 18)

% 12--

tan2 \1 \2–( )

tan2 \1 \2+( )--------------------------------

sin2 \1 \2–( )

sin2 \1 \2+( )--------------------------------+=

EE

n1n2----- \cos 0cos–

n1n2----- \cos 0cos+------------------------------------=

EM

EM

n1n2----- 0 \cos–cos

n1n2----- 0 \cos+cos------------------------------------=

\ 0 n1 n2

0.5 EE2 EM

2+( )

1 0.5 EE2 EM

2+( )–

Radiation Theory: Rosseland Model


No absorption takes place at the interface, so the probability of transmission plus reflectionis always one. If the photon is transmitted, then the angle of refraction is determined bySnells’ law:

(Eqn. 19)

ANSYS CFX performs these calculations at every radiation element boundary, although, inmost 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 originalenergy transport equation with a strongly temperature-dependent diffusion coefficient.

A good source for the simplification of the Radiation Transport Equation for the opticallythick limit can be seen in Siegel and Howe [23]. The total radiative heat flux in an opticallythick, 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, theconduction 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.

0sin\sin-----------

n2n1-----=

qr4

3' CKs–----------------------CEbv vd0

F7–=

'

q qc qr+=

k kr+( )CT–=

where kr16,n2T3

3'----------------------–=

k kr

Radiation Theory: The P1 Model


Wall Treatment

The Rosseland approximation is not valid near walls. Therefore, a special boundarycondition 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 TransportEquation, which assumes that the radiation intensity is isotropic or direction independentat a given location in space. The full form of the radiant energy equation and the derivationof the P1 model for radiation are given in Raithby [8]. Only a brief summary will be givenhere.

The spectral radiative heat flux in the diffusion limit for an emitting, absorbing, and linearlyscattering medium, can be computed as:

(Eqn. 27)

qr w,

qr w,, Tw

4 T g4–( )–

U-------------------------------=

U Tw T g

U

U 34G------

10v-----atan 0d

0

1

7=

0v

0v1G---

'k

803,T3w

--------------------- 20---–

1 0–1 0+------------ln–=

qrv1

3 K av Ksv–( ) AK sv–-------------------------------------------------- GEC–=

Radiation Theory: Discrete Transfer Model


The equation for the spectral incident radiation that results from substituting the aboveterms 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 directionallyindependent, 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 thescattering 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–-------------------------------------------------- GvC2 34 5C•– Kav Ebv Gv–( )=

A

n q: rv1

3 K av K sv–( ) AK sv–--------------------------------------------------6Gv

6n+---------–"v

2 2 "v–( )--------------------- Ebv Gv–[ ]w

= =

n n+

w

dIE r s,( )ds--------------------- K aE K sE+( )IE r s,( )– K aIb E T,( )

K sE4G-------- IE r s',( ) ^' S+d

4G7+ +=

IE r( ) IE r dr+( )n qER r( ) qE

R r dr+( )n

IE

IE r s,( ) IEo K aE K sE+( )s–( )exp Ibv 1 Kas–( )exp–( ) K sEIv+ +=

IEo

IE

Radiation Theory: Monte Carlo Model


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, ortemperature dependency of radiation quantities is overcome by iteration.

Since the objective of thermal radiation modeling is to obtain the total volumetricabsorption and emission, additional calculations are still needed. For the Gray spectralmodel, the calculation is done once for a unique radiation intensity field. For the Multibandand Multigray/Weighted Sum of Gray Gases, the solution must be computed for eachspectral band/ gray gas and a final integration to obtain the total radiation quantities isrequired. Under the assumption of coherent radiation field, ie., the solution at a givenfrequency 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, thedistance traveled, you can obtain the mean total intensity.

By following a typical selection of photons and tallying, in each volume element, thedistance 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, thedistance 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 theemissivity, 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 notusually important for heat transfer calculations. Providing that the spectral (Multiband orMultigray) selection is done properly, the Monte Carlo tallying automatically integrates overthe spectrum.

I Gqr

Ka

Ir

qvR r

Radiation Theory: Spectral Models


Spectral Models

The radiation intensity field is also a function of the spectrum as shown in (Eqn. 1). In orderto make the spectral dependence tractable in conjunction with a flow calculation, ANSYSCFX supports three different models for the spectral dependency of the radiative transferequation: Gray, Multiband and Multigray/Weighted Sum of Gray Gases.

Gray

The Gray model assumes that all radiation quantities are nearly uniform throughout thespectrum, consequently the radiation intensity is the same for all frequencies. Then, thedependency of (Eqn. 1) on frequency can be dropped.

This implies that only one radiative transfer equation must be solved and that all totalradiation 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. Thesebands should span the thermal radiation section of the spectrum. It is assumed that thevalue at a given spectral band is represented by the spectral band midpoint value infrequency domain.

ANSYS CFX assumes that the main spectral variable is frequency, since it is independent ofthe material refractive index and it will facilitate the setup of multidomain problems. Otherspectral variables, such wavelength and wavenumber would be available for vacuum only.

Then, the radiative transfer equation is integrated within is spectral band and a modifiedRTE 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

dBEIBE r s,( )ds------------------------------- KaE KsE+( )BEIBE r s,( )– KaFBEIb T( )

KsE4G-------- BEIBE r s',( )# s s'•( ) ^' S+d

4G7

+ +=

BEIBE

FBE Eb E T,( ) EdE1

E2

7 Eb E T,( ) Ed0

E2

7 Eb E T,( ) Ed0

E1

7–= =

I r s,( ) BEIBE r s,( )1

N

V=



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-offoverhead.

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 asa function of temperature and pL, that is the product of the partial pressure and the pathlength. In the context of typical combustion systems, the dominant emitters of radiation arecarbon 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 theemissivity arises as the result of independent emission from a sufficient number of graygases:

(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 NI N

pL

"g agi 1 ekiLp––( )

i 1=

N g

V=

"g

pL Fb

agii 1=

N g

V 1=

"g pL

agi

N g agi

ki



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 spectraldata 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 multigreygas 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 theabsorption 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

"g agi T g( ) 1 eK i pH2O pCO2

+( )L––

i 1=

4

V=

agi T g

agi b1i 10 5– b2iT g+=

Ki pH2O pCO2+( ) K i pH2O pCO2

pCO+ +( ) K HCi pHC+b

pCO pHC

b1i b2i K i K HCi

N g

N g



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





Discretization and SolutionTheory

Introduction


• 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 underideal conditions. To obtain solutions for real flows, a numerical approach must be adoptedwhereby the equations are replaced by algebraic approximations which may be solvedusing a numerical method.

Discretization of the Governing Equations

This approach involves discretizing the spatial domain into finite control volumes using amesh. The governing equations are integrated over each control volume, such that therelevant quantity (mass, momentum, energy, etc.) is conserved in a discrete sense for eachcontrol volume.

Discretization and Solution Theory: Numerical Discretization


The figure below shows a typical mesh with unit depth (so that it is two-dimensional), onwhich 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 passivescalar, expressed in Cartesian coordinates:

(Eqn. 1)

(Eqn. 2)

(Eqn. 3)

Node

Element Face Center

Finite Volume surface

Element

6%6t------ x j6

6 %U j( )+ 0=

66t----- %U i( ) x j6

6 %U jU i( )+ 6P6xi--------– x j6

6 µeffU i6x j6---------

U j6xi6----------+

2 38 94 5

2 38 94 5

+=

66t----- %0( ) x j6

6 %U j0( )+ x j66 $eff

06x j6--------2 34 5

2 34 5 S0+=



These equations are integrated over a control volume, and Gauss’ Divergence Theorem isapplied to convert some volume integrals to surface integrals. If control volumes do notdeform in time, then the time derivatives can be moved outside of the volume integrals andthe 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 volumeintegrals represent source or accumulation terms, and the surface integrals represent thesummation of the fluxes. Note that changes to these equations due to control volumedeformation 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 acoupled system of linearized algebraic equations. This is done by converting each term intoa 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 byapproximating specific values in each sector and then integrating those values over allsectors that contribute to a control volume. Surface flow terms are converted into theirdiscrete form by first approximating fluxes at integration points, ipn, which are located at

ddt----- % Vd

V7 %U j n jd

s7+ 0=

ddt----- %U i Vd

V7 %U jU i n jd

s7+ P n jd

s7– µeff

U i6x j6---------

U j6xi6----------+

2 38 94 5

n jds7 SU i

VdV7+ +=

ddt----- %0 Vd

V7 %U j0 n jd

s7+ $eff

06x j6--------2 34 5 n jd

s7 S0 Vd

V7+=

n3

Integration Point

Element Face Center

Sectors

ip1

ip2ip3

n1 n2



the center of each surface segment in a 3D element surrounding the control volume. Flowsare then evaluated by integrating the fluxes over the surface segments that contribute to acontrol volume.

Many discrete approximations developed for CFD are based on series expansionapproximations of continuous functions (such as the Taylor series). The order-accuracy ofthe approximation is determined by the exponent on the mesh spacing or timestep factorof the largest term in the truncated part of the series expansion. This is often the first termexcluded from the approximation. Increasing the order-accuracy of an approximationgenerally implies that errors are reduced more quickly with mesh or timestep sizerefinement. Unfortunately, in addition to increasing the computational load, high-orderapproximations are also generally less robust (i.e., less numerically stable) than theirlow-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 overall the integration points of the control volume. Note that the First Order Backward Eulerscheme 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 areidentical for all transport equations. As discussed by Patankar [118], however, naïveco-located methods lead to a decoupled (checkerboard) pressure field. Rhie and Chow [2]

V % %o–Bt--------------2 3

4 5 %U jBn j( )ipipV+ 0=

V%U i %oU i

o–Bt------------------------------

2 38 94 5

mip U i( )ipipV+ PBni( )ip +

ipV=

µeffU i6x j6---------

U j6xi6----------+

2 38 94 5

Bn j2 38 94 5

ipipV SU i

V+

V %0 %o0o

–Bt------------------------2 3

4 5 mip0ipipV+ $eff

06x j6--------Bn j2 3

4 5ipip

V S0V+=

Bt Cn j

mip

mip %U jBn j( )ip=



proposed an alternative discretization for the mass flows to avoid the decoupling, and thisdiscretization was modified by Majumdar [119] to remove the dependence of thesteady-state solution on the timestep.

A similar strategy is adopted in ANSYS CFX. By applying a momentum-like equation to eachintegration point, the following expression for the advecting (mass-carrying) velocity ateach 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 theintegration point, is augmented by a high order pressure variation that scales with the meshspacing. 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 thepressure redistribution term. This term is usually significantly smaller than the average ofvertex velocities, especially as the mesh is refined to reasonable levels.

In some cases, the pressure redistribution term can produce apparently significant spuriousvelocity 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 ipp6xi6-------

ip

p6xi6-------

ip

–2 38 94 5

cip f ip U i ip,o U i ip,

o–( )–+=

f ipdip

1 cipdip–----------------------=

dipVA----–=

A

cip%Bt------=

f ipp6xi6-------

ip

p6xi6-------

ip

–2 38 94 5

Bx( )3

Si



lead to velocity wiggles at locations where the body force is discontinuous (e.g., at a freesurface interface, or at the boundary of a porous region). The wiggles are greatly reduced oreliminated 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 arerespectively approximated using the old and current time level solution values. Theresulting discretization is:

(Eqn. 17)

It is robust, fully implicit, bounded, conservative in time, and does not yield a timestep sizelimitation. This discretization is, however, only first-order accurate in time and will introducediscretization errors that tend to diffuse steep temporal gradients. This behavior is similarto the numerical diffusion experienced with the Upwind Difference Scheme for discretizingthe advection term.

With the Second Order Backward Euler scheme, the start and end of timestep values arerespectively approximated as:

(Eqn. 18)

(Eqn. 19)

f ipp6xi6------- Si–2 3

4 5ip

p6xi6------- Si–2 3

4 5ip

–2 34 5

t66 %0 Vd

V7 V %0( )

n 12--+ %0( )

n 12--––

Bt------------------------------------------------d

t66 %0 Vd

V7 V %0 %o0o–

Bt------------------------2 34 5=

%0( )n 1

2--– %0( )o 12-- %0( )o %0( )oo–( )+=

%0( )n 1

2--+ %0( ) 12-- %0( ) %0( )o–( )+=



When these values are substituted into the general discrete approximation, (Eqn. 16), theresulting discretization is:

(Eqn. 20)

This scheme is also robust, implicit, conservative in time, and does not create a timesteplimitation. It is second-order accurate in time, but is not bounded and may create somenonphysical solution oscillations. For quantities such as volume fractions, whereboundedness is important, a modified Second Order Backward Euler scheme is usedinstead.

Shape Functions The solution fields are stored at the mesh nodes. However, various terms in the equationsrequire solutions or solution gradients to be approximated at integration points. Finiteelement shape functions are consequently used to evaluate the solution and its variationwithin 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. Theyare used to calculate various geometric quantities as well, including ip coordinates andsurface 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:

t66 %0 Vd

V7 V 1

Bt------32-- %0( ) 2 %0( )o– 1

2-- %0( )oo+2 34 5d

0

0 Ni0ii 1=

N node

V=

0i 0

Nii 1=

N node

V 1=

At node j, Ni1 i j=0 i jf;

=?

=

y Ni yii 1=

N node

V=



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=



Tetrahedral ElementFigure 4


(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




(Eqn. 27)

Pyramid ElementFigure 6


(Eqn. 28)

At this point, it is worth highlighting that these shape functions yield linear combinations ofpositively and negatively weighted nodal values. The significance of these positive andnegative 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=



Diffusion Terms Following the standard finite element approach, shape functions are used to evaluatespatial derivatives for all the diffusion terms. For example, for a derivative in the x directionat integration point ip:

(Eqn. 29)

The summation is over all the shape functions for the element. The Cartesian derivatives ofthe shape functions can be expressed in terms of their local derivatives via the Jacobiantransformation matrix:

(Eqn. 30)

The shape function gradients can be evaluated at the actual location of each integrationpoint (i.e., true tri-linear interpolation), or at the location where each ip surface intersects theelement edge (i.e., linear-linear interpolation). The latter formulation improves solutionrobustness at the expense of locally reducing the spatial order-accuracy of the discreteapproximation.

PressureGradient Term

The surface integration of the pressure gradient in the momentum equations involvesevaluating 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 theactual location of each integration point (i.e., true tri-linear interpolation), or at the locationwhere each ip surface intersects the element edge (i.e., linear-linear interpolation). As in thediscretization of the diffusion terms, the formulation improves solution robustness butreduces the local spatial order-accuracy of the discrete approximation.

606x------ ip

6Nn6x---------- ip

0nnV=

6N6x-------

6N6y-------

6N6z-------

6x6s------

6y6s------

6z6s-----

6x6t------

6y6t------

6z6t-----

6x6u------

6y6u------

6z6u------

1–6N6s-------

6N6t-------

6N6u-------

=

PBnip( )ip

Pip Nn sip tip uip, ,( )PnnV=



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 spatialgradients 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

0up

0

0ip 0up ' 0C Br:+=

0up r

C0C0

' C0

' 0=

' C0

'C0 Br–' 1=



reproduce steep spatial gradients. However, it will introduce dispersive discretization errorsthat tend to cause non-physical oscillations in regions of rapid solution variation as shownbelow. 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 andundesirable attributes of the Numerical Advection Correction Scheme. An additionalundesirable attribute is that this scheme may suffer from serious decoupling issues. Whileuse of this scheme is not generally recommended, it has proven both useful and accuratewhen 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, whichdepends upon pressure and the advecting velocity. For compressible flows, thediscretization of these terms is made as implicit as possible in each timestep with the use ofthe following Newton-Raphson linearization:

(Eqn. 35)

' C00ip

0ip Nn sip tip uip, ,( )0nnV=

'C0

'

'

%U( )nA %nU oA %oU nA %oU oA–+d



Here, the superscripts k and k-1 respectively indicate the current and previous iterates. Thisresults in an active linearization involving both the new density and velocity terms forcompressible flows at any Mach number.

MeshDeformation

The integral conservation equations presented above (Eqn. 4) must be modified when thecontrol volumes deform in time. These modifications follow from the application of theLeibnitz Rule:

(Eqn. 36)

where is the velocity of the control volume boundary.

As before, the differential conservation equations are integrated over a given controlvolume. At this juncture, the Leibnitz Rule is applied, and the integral conservationequations become:

(Eqn. 37)

(Eqn. 38)

(Eqn. 39)

The transient term accounts for the rate of change of storage in the deforming controlvolume, and the advection term accounts for the net advective transport across the controlvolume'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 thatfor each control volume, the rate of change of volume must exactly balance the net volumeswept due to the motion of its boundaries. The GCL is satisfied by using the same volumerecipes for both the control volume and swept volume calculations, rather than byapproximating the swept volumes using the mesh velocities.

tdd 0 Vd

V t( )7 06

t6------ VdV7 0W j n jd

s7+=

W j

tdd % Vd

V t( )7 % U j W j–( ) n jd

s7+ 0=

ddt----- %U i Vd

V t( )7 % U j W j–( )U i n jd

s7+ P n jd

s7– µeff

U i6x j6---------

U j6xi6----------+

2 38 94 5

n jds7 SU i

VdV7+ +=

ddt----- %0 Vd

V t( )7 % U j W j–( )0 n jd

s7+ $eff

06x j6--------2 34 5 n jd

s7 S0 Vd

V7+=

tdd Vd

V t( )7 W j n jd

s7=



The Coupled System of Equations

The linear set of equations that arise by applying the Finite Volume Method to all elementsin the domain are discrete conservation equations. The system of equations can be writtenin 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. Thenode may have any number of such neighbors, so that the method is equally applicable toboth structured and unstructured meshes. The set of these, for all control volumesconstitutes 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 beexpressed 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 ofthe rows of the matrix treated any differently (e.g., different solution algorithms formomentum versus mass). The advantages of such a coupled treatment over a non-coupled

ainb0i

nb

nbi

V bi=

0

0inb

ainb

auu auv auw aup

avu avv avw avp

awu awv aww awp

apu apv apw app i

nb

=

0inb

uvwp

nb

i

=

bi

bu

bv

bw

bp i

=

Discretization and Solution Theory: Solution Strategy - The Coupled Solver


or segregated approach are several: robustness, efficiency, generality and simplicity. Theseadvantages all combine to make the coupled solver an extremely powerful feature of anyCFD 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 firstsolved, 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 iterationsare typically required in addition to the need for judiciously selecting relaxation parametersfor 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 equationsat any given timestep. For steady state problems, the time-step behaves like an ‘accelerationparameter’, to guide the approximate solutions in a physically based manner to asteady-state solution. This reduces the number of iterations required for convergence to asteady 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 ANSYSCFX-Solver.

The solution of each set of field equations shown in the flow chart consists of twonumerically 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.



When solving fields in the ANSYS CFX-Solver, the outer- or timestep-iteration is controlledby the physical time scale or timestep for steady and transient analyses, respectively. Onlyone 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) factorizationtechnique for solving the discrete system of linearized equations. It is an iterative solverwhereby the exact solution of the equations is approached during the course of severaliterations.

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

YesYes

Advance inTime



The linearized system of discrete equations described above can be written in the generalmatrix 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 thenumber of computational mesh elements increases. Performance also tends to rapidlydecrease if there are large element aspect ratios present. The performance of the solver canbe greatly improved by employing a technique called ‘multigrid.’

AlgebraicMultigrid

The convergence behavior of many matrix inversion techniques can be greatly enhanced bythe use of a technique called ‘multigrid’. The multigrid process involves carrying out earlyiterations on a fine mesh and later iterations on progressively coarser virtual ones. Theresults 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 agiven mesh size, iterative solvers are only efficient at reducing errors which have awavelength of the order of the mesh spacing. So, while shorter wavelength errors disappearquite quickly, errors with longer wavelengths, of the order of the domain size, can take anextremely long time to disappear. The Multigrid Method bypasses this problem by using aseries of coarse meshes such that longer wavelength errors appear as shorter wavelengtherrors relative to the mesh spacing. To prevent the need to mesh the geometry using aseries of different mesh spacings, ANSYS CFX uses Algebraic Multigrid.

Algebraic Multigrid [25] forms a system of discrete equations for a coarse mesh by summingthe fine mesh equations. This results in virtual coarsening of the mesh spacing during thecourse of the iterations, and then re-refining the mesh to obtain an accurate solution. This

A 0 b=

0

0'

0n 1+ 0n 0'+=

0'

A0' rn=

rn b A0n–=



technique significantly improves the convergence rates. Algebraic Multigrid is lessexpensive than other multigrid methods since discretization of the non-linear equations isonly performed once for the finest mesh.

ANSYS CFX uses a particular implementation of Algebraic Multigrid called AdditiveCorrection [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 thebalance of conserved quantities over a control volume. The coarse mesh equations can becreated by merging the original control volumes to create larger ones as shown below. Thediagram shows the merged coarse control volume meshes to be regular, but in general theirshape becomes very irregular. The coarse mesh equations thus impose conservationrequirements over a larger volume and in so doing reduce the error components at longerwavelengths.

Original mesh

First coarse mesh (virtual)

Next coarse mesh (virtual)

Discretization and Solution Theory: Discretization Errors


Residual Normalization Procedure

As described above, the raw residual, [r], is calculated as the imbalance in the linearizedsystem of discrete equation. The raw residuals are then normalized for the purpose ofsolution monitoring and to obtain a convergence criteria. An overview of the normalizationprocedure 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 inlocations where the volume fraction is negligible having a large influence.

Discretization Errors

There are often differences between the exact analytical solution of the modeled differentialequations (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 bylocalized sources and propagated (i.e., amplified, advected, diffused) throughout thesolution domain. Localized sources of error result from the high-order terms that areexcluded 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 discreteapproximations. Both error sources and propagation are affected by the solution and meshdistributions, as discussed in Controlling Error Sources (p. 296) and Controlling ErrorPropagation (p. 297).

Controlling Error Sources

Reducing the source of solution error (i.e., the magnitude of terms excluded in the discreteapproximations) is critical if accurate numerical solutions are desired. The two mosteffective strategies of accomplishing this are by increasing the order-accuracy of discreteapproximations (e.g., using the high resolution rather than the upwind difference advection

0

r0[ ]r0[ ]

apB0-------------=

r0B0

B0



scheme) and/or by reducing the mesh spacing in regions of rapid solution variation. Theformer strategy is discussed above (see Advection Term (p. 288)), and implications of thelatter are now considered.

Finely spaced isotropic mesh distributions are ideal, but they are often not tractable. A moreuseful strategy for reducing sources of solution error is to generate anisotropic meshes withfine spacing in directions of most rapid solution variation and relatively coarse spacing inother directions. This is exemplified by typical boundary layer meshes that are compressedin 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 poorgeometrical mesh quality (see Measures of Mesh Quality (p. 371 in "ANSYS CFX-SolverModeling Guide")). In particular, the error source contributions due to the discretization oftransient/storage, diffusion, source and Rhie-Chow redistribution terms increase with meshanisotropy. This is why, for example, high orthogonality and low expansion factors arerecommended 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 canbe 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 oferror is more easily controlled.

Errors are amplified by strong unphysical influences in the discrete form of the modeledequations. These unphysical influences will lead to convergence difficulties, and in extremecases, complete divergence. Similar to error sources, error amplification is controlledthrough the choice of discretization and/or the mesh distribution.

As highlighted above, (unphysical) negative influences may be introduced into thediscretization through the finite-element shape functions, and these influences may growas geometrical mesh quality (see Measures of Mesh Quality (p. 371 in "ANSYS CFX-SolverModeling Guide")) deteriorates. Tri-linear shape functions, for example, are moresusceptible to negative influences than linear-linear shape functions. Nevertheless, tri-linearshape functions are used as much as possible due to their improved accuracy, andlinear-linear shape functions are used whenever solution robustness is critical.

Geometrical mesh quality is important regardless of the shape functions used. An excellentexample 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 flowterms is rendered physically invalid. The amount of a conserved quantity within a controlvolume will actually decrease given a net flow into the control volume.



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

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