CFD-ACE V2009.0 Modules Manual Part1

ESI CFD Inc. 6767 Old Madison Pike, Ste. 600

Huntsville, AL 35806 Phone: (256) 713-4700 Fax: (256) 713-4799

Software Support: [email protected] Software Sales: [email protected]

CFD-ACE+ V2009.0 Modules Manual Part 1

mailto:[email protected]


Table Of Contents About ESI GROUP .......................................................................................................................... 1

Copyright Information................................................................................................................... 1 About CFD-ACE+ ............................................................................................................................ 3 New Features and Improvements - V2009.0................................................................................... 6

CFD-ACE-SOVLER - V2009.0.1.................................................................................................. 6 Fast Time Stepping (FTS) Enhancements............................................................................... 6 Force and Moment Coefficients on Specified Surfaces ........................................................... 6 Forces and Moments for Momentum Resistance .................................................................... 7 V2F Model Enhancement......................................................................................................... 7 Two-Phase Model Improvement .............................................................................................. 8 Two Phase Porous Media ........................................................................................................ 8

CFD-ACE-GUI - V2009.0.1.......................................................................................................... 9 ESI Mobile Support .................................................................................................................. 9

Using Help ..................................................................................................................................... 10 Getting Started............................................................................................................................... 11

How to execute the software: ................................................................................................. 11 How to add shortcuts to the Start Menu:................................................................................ 11

How to Report Problems ............................................................................................................... 13 Flow Module .................................................................................................................................. 14

Flow Module Introduction to Flow Module ................................................................................. 14 Flow Module Applications of the Flow Module........................................................................... 14

Flow Visualization................................................................................................................... 14 Pressure Field Calculations.................................................................................................... 14 Mass Flow Calculations.......................................................................................................... 14 Multi-Physics Applications...................................................................................................... 14

Flow Module Features and Limitations of the Flow Module ...................................................... 15 Features ................................................................................................................................. 15

Non-Newtonian Viscosity Options ...................................................................................... 15 Swirl Model ......................................................................................................................... 15 Slip Wall Boundary Conditions ........................................................................................... 15 Hemolysis Model ................................................................................................................ 15 Simple Flow Models............................................................................................................ 16

Limitations .............................................................................................................................. 16 Theory ........................................................................................................................................ 16

Flow Module Flow Module Theory ......................................................................................... 16 Mass Conservation[1] .......................................................................................................... 16 Momentum Conservation[1] ................................................................................................. 17 Navier-Stokes Equations[1].................................................................................................. 17

Flow Module Simple Flow Model Theory ............................................................................... 19 Second Order Wall Model................................................................................................... 19 One Cell Wall Model ........................................................................................................... 19

Flow Module Slip Wall Theory................................................................................................ 20 Model Setup............................................................................................................................... 21

Flow Module Implementation and Grid Generation................................................................ 21 Flow Module Problem Type.................................................................................................... 22 Model Options ........................................................................................................................ 22

Flow Module Model Options - Shared Tab ......................................................................... 22 Flow Module Model Options - Flow Tab ............................................................................. 22 Flow Module Model Options - Advanced Tab..................................................................... 23

Flow Module Volume Conditions............................................................................................ 24 Fluid Properties ...................................................................................................................... 25

Density ................................................................................................................................ 25 Viscosity.............................................................................................................................. 27

ii

Table Of Contents

Boundary Conditions .............................................................................................................. 33 Flow Module Boundary Conditions..................................................................................... 33 Flow Module Boundary Conditions - Inlets ......................................................................... 33 Flow Module Boundary Conditions - Outlets ...................................................................... 36 Flow Module Boundary Conditions - Walls and Rotating Walls ......................................... 38 Flow Module Boundary Conditions - Symmetry ................................................................. 39 Flow Module Boundary Conditions - Interfaces.................................................................. 39 Flow Module Boundary Conditions - Thin Walls................................................................. 39 Flow Module Boundary Conditions - Cyclic/Periodic.......................................................... 39

Flow Module Initial Conditions................................................................................................ 39 Flow Module Solver Control Settings ..................................................................................... 40

Spatial Differencing Tab ..................................................................................................... 40 Solver Selection.................................................................................................................. 40 Relaxation Parameters ....................................................................................................... 40 Variable Limits .................................................................................................................... 40 Advanced Settings.............................................................................................................. 41

Flow Module Output Options.................................................................................................. 41 Output ................................................................................................................................. 41 Printed Output..................................................................................................................... 42 Graphical Output................................................................................................................. 42

Flow Module Post Processing.................................................................................................... 42 Flow Module Frequently Asked Questions ................................................................................ 44 Flow Module Examples.............................................................................................................. 53 Flow Module References ........................................................................................................... 53

Heat Transfer Module.................................................................................................................... 55 Heat Transfer Module Introduction to the Heat Module............................................................. 55 Heat Transfer Module Applications............................................................................................ 55

Thermal Field Calculations..................................................................................................... 55 Heat Transfer Calculations..................................................................................................... 55 Pure Conduction Problems .................................................................................................... 55 Conjugate Heat Transfer Problems........................................................................................ 56 Natural Convection Problems................................................................................................. 56 Multi-Physics Applications...................................................................................................... 56

Heat Transfer Module Features and Limitations of the Heat Module........................................ 56 Features ................................................................................................................................. 56

Ice Melting........................................................................................................................... 56 Solidification........................................................................................................................ 56 Moving Solids ..................................................................................................................... 57 Wall Heat Sources .............................................................................................................. 57

Limitations .............................................................................................................................. 57 Heat Transfer Module Theory .................................................................................................... 57 Heat Module Model Setup ......................................................................................................... 58

Heat Transfer Module Implementation and Grid Generation ................................................. 58 Heat Transfer Module Problem Type ..................................................................................... 58 Heat Transfer Module Model Options-Shared Tab ................................................................ 59 Heat Transfer Module Model Options-Heat Tab .................................................................... 59

Ice Melting........................................................................................................................... 59 Solidification........................................................................................................................ 59 Moving Solid ....................................................................................................................... 59

Heat Module Volume Conditions............................................................................................ 60 Heat Transfer Module Volume Conditions.......................................................................... 60 Heat Transfer Module Volume Conditions-Ice Melting Properties ..................................... 65 Heat Transfer Module Volume Conditions-Solidification Properties .................................. 65 Heat Transfer Module Volume Conditions-Moving Solid Properties .................................. 65

Boundary Conditions .............................................................................................................. 66 Heat Transfer Module Boundary Conditions-Introduction .................................................. 66

iii

CFD-ACE_V2009.0_Modules_Manual_Part1

Heat Transfer Module Boundary Conditions-Inlets/Outlets ................................................ 66 Heat Transfer Module Boundary Conditions-Walls/Rotating Walls .................................... 66 Heat Transfer Module Boundary Conditions-Symmetry..................................................... 70 Heat Transfer Module Boundary Conditions-Interfaces ..................................................... 71 Heat Transfer Module Boundary Conditions-Thin Walls .................................................... 71 Heat Transfer Module Boundary Conditions-Cyclic ........................................................... 71

Heat Transfer Module Initial Conditions ................................................................................. 71 Heat Transfer Module Solver Control Settings....................................................................... 71

Spatial Differencing Scheme .............................................................................................. 71 Solver Selection.................................................................................................................. 72 Relaxation Parameters ....................................................................................................... 72 Variable Limits .................................................................................................................... 72 Advanced Settings.............................................................................................................. 72

Heat Transfer Module Output Options ................................................................................... 73 Output ................................................................................................................................. 73 Printed Output..................................................................................................................... 73 Graphical Output................................................................................................................. 74

Heat Transfer Module Post Processing ..................................................................................... 74 Heat Module Frequently Asked Questions ................................................................................ 75 Heat Transfer Module Examples ............................................................................................... 76 Heat Transfer Module References............................................................................................. 76

Turbulence Module........................................................................................................................ 77 Turbulence Module Introduction ................................................................................................ 77 Turbulence Module Applications................................................................................................ 77 Turbulence Module Features ..................................................................................................... 77 Turbulence-Theory..................................................................................................................... 78

Turbulence Module Theory-Introduction ................................................................................ 78 Reynolds Averaged Navier-Stokes Simulations..................................................................... 79

Turbulence Module Theory-Standard k- Model ................................................................ 79 Turbulence Module Theory-RNG k- Model ....................................................................... 81 Turbulence Module Theory-Kato-Launder k- Model ......................................................... 82 Turbulence Module Theory-Low Reynolds Number k- Model (Chien) ............................. 82 Turbulence Module Theory-Two-Layer k- Model .............................................................. 83 Turbulence Module Theory-k- Model ............................................................................... 84 Turbulence Module Theory-k- SST Model ....................................................................... 85 Turbulence Module Theory-Spalart-Allmaras Model.......................................................... 87 Turbulence Module Theory - V2F Model ............................................................................ 88 Turbulence Module Theory - Modified V2F Model ............................................................. 89

Large Eddy Simulations ......................................................................................................... 90 Turbulence Module Theory-Large Eddy Simulations Introduction ..................................... 90 Turbulence Module Theory-Large Eddy Simulations-SGS Models Introduction................ 90 Turbulence Module Theory-Large Eddy Simulations-SGS Models-Smagorinsky Model... 90 Turbulence Module Theory-Large Eddy Simulations-SGS Models-Germano's Dynamic Subgrid-Scale Model .......................................................................................................... 91 Turbulence Module Theory-Large Eddy Simulations-SGS Models-Menon's Localized Dynamic Subgrid-Scale Model (LDKM).............................................................................. 91

Transition Models ................................................................................................................... 93 Turbulence Theory Transition Models Introduction ............................................................ 93 Turbulence Theory Transport Equation for Intermittency................................................... 93 Turbulence Theory Transport Equation for Transition Momentum Thickness Reynolds Number ............................................................................................................................... 95 Turbulence Theory Correction for Separation Induced Transition ..................................... 96 Turbulence Theory Coupling with Turbulence Model......................................................... 97 Transition Models-Empirical Correlations........................................................................... 98

Turbulence Module Constant Turbulent Viscosity................................................................ 100

iv

Table Of Contents

Turbulence Module User Defined Turbulent Viscosity......................................................... 100 Turbulence Module Turbulence in Porous Media ................................................................ 100 Turbulence Module Wall Functions...................................................................................... 101

Non-Equilibrium Model (Pressure gradient sensitize velocity log-law)............................. 101 Two Layer Model .............................................................................................................. 103

Turbulence Module Limitations................................................................................................ 104 Turbulence-Implementation ..................................................................................................... 104

Turbulence Module Implementation Introduction................................................................. 104 Turbulence Module Implementation-Grid Generation.......................................................... 104

Grid Parameter R.............................................................................................................. 104 Model Setup and Solution .................................................................................................... 105

Turbulence Module Implementation-Model Setup and Solution-Introduction .................. 105 Turbulence Module Implementation-Model Setup and Solution-Problem Type............... 105 Model Options................................................................................................................... 105 Turbulence Module Implementation-Model Setup and Solution-Volume Conditions ....... 107 Boundary Conditions ........................................................................................................ 107 Turbulence Module Implementation-Model Setup and Solution-Initial Conditions........... 113 Model Setup and Solution-Solver Control ........................................................................ 113 Model Setup and Solution-Output .................................................................................... 116

Turbulence Module Implementation-Post Processing ......................................................... 118 Turbulence Module Frequently Asked Questions.................................................................... 118 Turbulence Module Examples ................................................................................................. 119 Turbulence Module References............................................................................................... 119

Chemistry Module........................................................................................................................ 121 Chemistry Module Introduction ................................................................................................ 121 Chemistry-Applications ............................................................................................................ 121

Chemistry Module Applications-Introduction........................................................................ 121 Chemistry Module Applications-Mixing Only........................................................................ 122 Chemistry Module Applications-Mixing with Gas Phase Reactions..................................... 122 Chemistry Module Applications-Mixing with Surface Reactions .......................................... 122 Chemistry Module Applications-Multi-Physics Applications................................................. 122

Chemistry-Features ................................................................................................................. 122 Chemistry Module Features Introduction ............................................................................. 122 Chemistry Module Features-Solution Approach .................................................................. 123

Mixture Mass Fractions..................................................................................................... 123 Species Mass Fractions.................................................................................................... 123

Chemistry Module Features-Mass Diffusion Options........................................................... 123 Features-Gas Phase Reactions ........................................................................................... 123

Chemistry Module Features-Gas Phase Reactions-Introduction ..................................... 123 Chemistry Module Features-Gas Phase Reactions-Instantaneous Reaction Model ....... 124 Chemistry Module Features-Gas Phase Reactions-Equilibrium Reaction Model ............ 124 Chemistry Module Features-Gas Phase Reactions-Finite-Rate Model (for Mixture Solution).......................................................................................................................................... 124 Chemistry Module Features-Gas Phase Reactions-Finite-Rate Model (for Species Solution)............................................................................................................................ 124 Chemistry Module Features-Gas Phase Reactions-Eddy Breakup Model ...................... 125 Chemistry Module Features-Gas Phase Reactions-Prescribed PDF Model.................... 125

Chemistry Module Features-Surface Reactions .................................................................. 125 Chemistry Module Features-Coupled Solver ....................................................................... 125 Chemistry Module Features-Unsteady Combustion ............................................................ 125

Chemistry-Theory .................................................................................................................... 126 Chemistry Module Theory Introduction ................................................................................ 126 Chemistry-Theory-Definitions And Relations ....................................................................... 126

Chemistry Module Theory-Definitions and Relations-Introduction ................................... 126 Chemistry Module Theory-Definitions and Relations-Composition Variables.................. 126 Chemistry Module Theory-Definitions and Relations-Chemical Rate Expressions.......... 127

v


Chemistry Module Theory-Definitions and Relations-Mixture Fractions .......................... 131 Chemistry-Theory-Gas Phase Reaction Models.................................................................. 132

Chemistry Module Theory-Gas Phase Reaction Models-Introduction ............................. 132 Chemistry Module Theory-Gas Phase Reaction Models-Instantaneous Chemistry Model.......................................................................................................................................... 132 Chemistry Module Theory-Gas Phase Reaction Models-Equilibrium Model ................... 133 Chemistry Module Theory-Gas Phase Reaction Models-Finite-Rate Model (for Mixture Solution)............................................................................................................................ 135 Chemistry Module Theory-Gas Phase Reaction Models-Finite-Rate Model (for Species Solution)............................................................................................................................ 136

Chemistry Module Theory-Surface Reaction Models .............................................................. 137 Chemistry-Turbulence-Combustion Interaction ....................................................................... 140

Chemistry Module Theory-Turbulence-Combustion Interaction-Introduction ...................... 140 Turbulence-Combustion Interaction-Determining PDF ........................................................ 141

Chemistry Module Theory-Turbulence-Combustion Interaction-Determining PDF.......... 141 Chemistry Module Theory-Turbulence-Combustion Interaction-Reaction Progress PDF 142 Chemistry Module Theory-Turbulence-Combustion Interaction-Mixture Fraction PDF ... 143

Chemistry Module Turbulence-Combustion Interaction-Determining Averaged Variables.. 146 Chemistry Module Turbulence-Combustion Interaction-Operator Splitting ......................... 146 Chemistry Module Turbulence-Combustion Interaction-In Situ Adaptive Tabulation (ISAT)147 Chemistry Module Turbulence-Combustion Interaction-Subgrid Linear Eddy Model .......... 149 Chemistry Module Turbulence-Combustion Interaction-Application to Large Eddy Simulation.............................................................................................................................................. 151

Chemistry Module Limitations.................................................................................................. 152 Chemistry-Implementation ....................................................................................................... 152

Chemistry Module Implementation-Introduction................................................................... 153 Chemistry Module Implementation-Grid Generation............................................................ 153 Implementation-Model Setup and Solution .......................................................................... 153

Chemistry Module Model Setup and Solution-Introduction .............................................. 153 Chemistry Module Implementation-Model Setup and Solution-Problem Type................. 153 Model Setup and Solution-Model Options ........................................................................ 153 Model Setup and Solution-Volume Conditions................................................................. 161 Model Setup and Solution-Boundary Conditions.............................................................. 164 Chemistry Module Implementation-Model Setup and Solution-Initial Conditions............. 166 Model Setup and Solution-Solver Control ........................................................................ 167 Model Setup and Solution-Output .................................................................................... 168

Chemistry Module Implementation-Post Processing ........................................................... 170 Chemistry Module Frequently Asked Questions...................................................................... 171 Chemistry Module Examples ................................................................................................... 172 Chemistry Module References................................................................................................. 172

User Scalar Module ..................................................................................................................... 173 User Scalar Module Introduction.............................................................................................. 173 User Scalar Module Applications............................................................................................. 173 User Scalar Module Features .................................................................................................. 173

Scalar Types......................................................................................................................... 173 Scalar Control....................................................................................................................... 174

User Scalar Module Theory ..................................................................................................... 174 User Scalar Module Limitations ............................................................................................... 175 User Scalar-Implementation .................................................................................................... 175

User Scalar Module Implementation-Introduction................................................................ 175 User Scalar Module Implementation-Grid Generation ......................................................... 175 Implementation-Model Setup and Solution .......................................................................... 175

User Scalar Module Implementation-Model Setup and Solution-Introduction.................. 175 User Scalar Module Implementation-Model Setup and Solution-Problem Type .............. 176 User Scalar Module Implementation-Model Setup and Solution-Model Options ............. 176 User Scalar Module Implementation-Model Setup and Solution-Volume Conditions ...... 176

vi

Table Of Contents

User Scalar Module Implementation-Model Setup and Solution-Boundary Conditions... 177 User Scalar Module Implementation-Model Setup and Solution-Initial Conditions .......... 179 Model Setup and Solution-Solver Control ........................................................................ 179 User Scalar Module Implementation-Model Setup and Solution-Output.......................... 181

User Scalar Module Implementation-Post Processing......................................................... 182 User Scalar Module Frequently Asked Questions................................................................... 183 User Scalar Module References.............................................................................................. 183

Radiation Module......................................................................................................................... 184 Radiation Module Introduction ................................................................................................. 184 Radiation Module Applications................................................................................................. 184 Radiation Module Features...................................................................................................... 185 Radiation-Theory ..................................................................................................................... 185

Radiation Module Theory-Blackbody Radiation................................................................... 185 Radiation Module Theory-Radiation Properties ................................................................... 187 Radiation Module Theory-Radiation Characteristics of Gases ............................................ 188 Radiation Module Theory-Radiative Transfer Equation (RTE) ............................................ 188 Theory-Solution Method ....................................................................................................... 189

Radiation Module Theory-Solution Method Introduction .................................................. 189 Radiation Module Theory-Solution Method-Surface-to-Surface Method ......................... 190 Solution Method-Discrete Ordinate Method ..................................................................... 191 Solution Method-Monte Carlo Method.............................................................................. 194 Radiation Module Theory-Solution Method-P1 Method ................................................... 196

Radiation Module Limitations................................................................................................... 198 Radiation-Implementation ........................................................................................................ 199

Radiation Module Implementation-Introduction.................................................................... 199 Radiation Module Implementation-Grid Generation............................................................. 199 Implementation-Model Setup and Solution .......................................................................... 199

Radiation Module Implementation-Model Setup and Solution-Introduction ..................... 199 Radiation Module Implementation-Model Setup and Solution-MC Model Requirements 199 Radiation Module Implementation-Model Setup and Solution-Problem Type.................. 200 Model Setup and Solution-Model Options ........................................................................ 200 Model Setup and Solution-Data........................................................................................ 201 Radiation Module Implementation-Model Setup and Solution-Volume Conditions-Introduction ....................................................................................................................... 211 Model Setup and Solution-Boundary Conditions.............................................................. 212 Radiation Module Implementation-Model Setup and Solution-Initial Conditions.............. 213 Radiation Module Implementation-Model Setup and Solution-Solver Control ................. 214 Radiation Module Implementation-Model Setup and Solution-Output ............................. 214

Radiation Module Implementation-Post Processing ............................................................ 214 Radiation Module Frequently Asked Questions....................................................................... 215 Radiation Module References.................................................................................................. 219

Cavitation Module........................................................................................................................ 220 Cavitation Module Introduction ................................................................................................ 220 Cavitation-Applications ............................................................................................................ 220

Cavitation Module Applications-Introduction ........................................................................ 220 Cavitation Module Applications-Automotive/Hydraulic Applications .................................... 221 Cavitation Module Applications-Turbomachinery Problems ................................................ 223 Cavitation Module Applications-Hydrofoil Problems ............................................................ 225

Cavitation Module Features ..................................................................................................... 226 Cavitation-Theory..................................................................................................................... 227

Cavitation Module Theory-Introduction ................................................................................ 227 Cavitation Module Theory-Vapor Transport Equations........................................................ 228 Cavitation Module Theory-Effect of Turbulence................................................................... 228 Cavitation Module Theory-Effect of Non-Condensable Gases ............................................ 229

Cavitation Module Limitations .................................................................................................. 230 Fluid Properties .................................................................................................................... 230

vii


Activating Cavitation............................................................................................................. 230 Isothermal Assumption......................................................................................................... 230 Modules Not Supported........................................................................................................ 230

Cavitation-Implementation ....................................................................................................... 230 Cavitation Module Implementation-Introduction................................................................... 230 Cavitation Module Implementation-Grid Generation............................................................ 230 Implementation-Model Setup and Solution .......................................................................... 231

Cavitation Module Implementation-Model Setup and Solution-Introduction .................... 231 Cavitation Module Implementation-Model Setup and Solution-Problem Type................. 231 Model Setup and Solution-Model Options ........................................................................ 231 Cavitation Module Implementation-Model Setup and Solution-Volume Conditions ......... 233 Cavitation Module Implementation-Model Setup and Solution-Boundary Conditions...... 234 Cavitation Module Implementation-Model Setup and Solution-Initial Conditions............. 234 Model Setup and Solution-Solver Control ........................................................................ 234 Cavitation Module Implementation-Model Setup and Solution-Output ............................ 235

Cavitation Module Implementation-Post Processing............................................................ 236 Cavitation Module Frequently Asked Questions...................................................................... 236 Cavitation Module References................................................................................................. 236

Grid Deformation Module ............................................................................................................ 238 Grid Deformation Module Introduction..................................................................................... 238 Grid Deformation-Applications ................................................................................................. 238

Grid Deformation Module Applications-Introduction............................................................. 238 Grid Deformation Module Applications-Fluid-Structures Interaction Problems.................... 238 Grid Deformation Module Applications-Simple Prescribed Motion ...................................... 238 Grid Deformation Module Applications-User Defined Motion .............................................. 238

Grid Deformation-Features ...................................................................................................... 239 Grid Deformation Module Features-Introduction.................................................................. 239 Grid Deformation Module Features-Automatic Remeshing ................................................. 239 Grid Deformation Module ..................................................................................................... 239 Features-Automatic Re-meshing.......................................................................................... 239 Transfinite Interpolation Scheme ......................................................................................... 239 Grid Deformation Module ..................................................................................................... 239 Features-Automatic Re-meshing.......................................................................................... 239 Solid-body Elasticity Analogy ............................................................................................... 239 Grid Deformation Module Features-User Defined Remeshing ............................................ 241

Grid Deformation Module Limitations ...................................................................................... 241 Grid Deformation-Implementation............................................................................................ 241

Grid Deformation Module Implementation-Introduction ....................................................... 242 Grid Deformation Module Implementation-Grid Generation ................................................ 242 Implementation-Model Setup and Solution .......................................................................... 242

Grid Deformation Module Implementation-Model Setup and Solution-Introduction ......... 242 Grid Deformation Module Implementation-Model Setup and Solution-Problem Type ..... 242 Model Setup and Solution-Model Options ........................................................................ 242 Grid Deformation Module Implementation-Volume Conditions ........................................ 243 Model Setup and Solution-Boundary Conditions.............................................................. 244

Grid Deformation Module Implementation-Specialized Point Constraint ............................. 246 Tips on Moving Grid Setup ............................................................................................... 246 Domain Division................................................................................................................ 246 Using the .spc File ............................................................................................................ 250

Grid Deformation Module Implementation-Post Processing ................................................ 253 Grid Deformation Module Frequently Asked Questions .......................................................... 253

Stress Module.............................................................................................................................. 254 Stress Module Introduction ...................................................................................................... 254 Stress-Applications .................................................................................................................. 254

Stress Module Applications Introduction .............................................................................. 254 Modules-Stress Module Application-Pure Structural Analysis ............................................. 254

viii

Table Of Contents

ix

Stress Module Applications-Coupled Solid/Fluid/Thermal Problems................................... 255 Stress Module Applications-Multi-disciplinary Electrostatic Problems-MEMS..................... 256

Stress Module Features........................................................................................................... 259 Stress-Theory........................................................................................................................... 260

Stress Module Theory-Introduction ...................................................................................... 260 Stress Module Theory-Damping........................................................................................... 262

Stress-Limitations .................................................................................................................... 265 Stress Module Limitations-Limited Element Library............................................................. 265 Stress Module Limitations-Arbitrary Interfaces .................................................................... 265 Stress Module Limitations-Cyclic/Thin Wall Boundary Conditions....................................... 265

Stress-Implementation ............................................................................................................. 265 Stress Module Implementation Introduction......................................................................... 265 Stress Module Implementation-Grid Generation.................................................................. 265

Element Types.................................................................................................................. 265 Choosing An Element Type.............................................................................................. 267 Structural Analysis ............................................................................................................ 268

Implementation-Model Setup and Solution .......................................................................... 269 Stress Module Implementation-Model Setup and Solution-Introduction .......................... 269 Stress Module Implementation-Model Solution and Setup-Problem Type....................... 269 Model Setup and Solution-Model Options ........................................................................ 270 Model Setup and Solution-Volume Conditions................................................................. 273 Model Setup and Solution-Boundary Conditions.............................................................. 278 Stress Module Implementation-Model Setup and Solution-Initial Conditions................... 287 Model Setup and Solution-Solver Control ........................................................................ 288 Model Setup and Solution-Output .................................................................................... 292

Stress Module Implementation-Post-Processing ................................................................. 298 Stress Module Frequently Asked Questions............................................................................ 299 Stress-Examples...................................................................................................................... 305

Stress Module Examples...................................................................................................... 305 Stress-Examples-Demo Problems ....................................................................................... 305

Stress Module Examples-Stress Concentration............................................................... 305 Stress Module Examples-Hoop Stress............................................................................. 307 Stress Module Examples-Large Deflection ...................................................................... 308

Stress-Examples-Validation Cases...................................................................................... 309 Stress Module Examples-Stress Concentration in a Circular Cylinder ............................ 309 Stress Module Examples-Thermoelastic Deformation of a Cylinder ................................ 310

Stress Module References....................................................................................................... 312 Appendix A - Post Processing Variables..................................................................................... 314

Appendix A Post-Processing Variables .................................................................................. 314 Index ............................................................................................................................................ 321

About ESI GROUP

ESI CFD is a technology leader in the field of advanced computational fluid dynamics simulation software backed by more than 20 years of research based knowledge throughout a wide range of industries. ESI CFD’s broad range of products and services provide all of the necessary tools for advanced multiphysics analysis in a virtual prototype environment, significantly reducing time and expense through comprehensive up-front modeling and simulation. Key focus areas include microfluidics, biomedical, plasma, MEMS, fuel cells, semiconductor, automotive and aerospace.

ESI CFD’s product portfolio represents a unique collaborative, virtual engineering solution, known as the Virtual Try-Out Space (VTOS), enabling a continuous improvement on the virtual prototype. By drastically reducing costs and development lead times, VTOS solutions offer major competitive advantages by progressively eliminating the need for physical prototypes.

Copyright Information

™1997-2008 by ESI-Group

1


2

All rights reserved. Published 2008.

This information is the confidential and proprietary product of ESI-Group. Any unauthorized use, reproduction, or transfer of this manual is strictly prohibited. Subject to limited distribution and restricted disclosure only.

CFD-ACE™, CFD-ACE+™, CFD-CADalyzer™, CFD-VIEW™, CFD-GEOM™, SimManager™, CFD-VisCART™, CFD-TOPOTM and CFD-FASTRAN™ are registered trademarks of ESI-Group.

Portions of this software are owned by Spatial Corp.

Copyright© 1989-2008 All rights reserved.

About CFD-ACE+

CFD-ACE+ is a set of computer applications for multi-physics computational analysis. The programs provide an integrated geometry and grid generation software, a graphical user interface for preparing the model, a computational solver for performing the simulation, and an interactive visualization software for examining and analyzing the simulation results.

The standard CFD-ACE+ package includes the following applications:

CFD-GEOM - geometry and grid generation

CFD-VisCART -

CFD-ACE-GUI - graphical user interface to the CFD-ACE-SOLVER

CFD-ACE-SOLVER - advanced, multiphysics solver

CFD-VIEW - interactive post processor

The information contained within specifically addresses the CFD-ACE-SOLVER and its interaction with CFD-ACE-GUI. A schematic representation of the applications is shown below.

3


Schematic Representation of CFD-ACE+

CFD-ACE+ provides an interactive tool kit for building the input required for the CFD-ACE-Solver. You can use it in conjunction with other ESI CFD products to form a complete solution analysis package. Other ESI CFD products include:

CFD-VisCART - provides Cartesian and viscous Cartesian grid generation capabilities.

CADalyzer - works with native CAD geometries and provides automatic grid generation for CFD calculations

4

Intro-About CFD-ACE+

5

CFD-TOPO - predicts the transport, chemistry, etch and deposition of semiconductor materials on the microscopic scales

SimManager - uses the CFD-ACE+ package to perform parametric and optimization studies using various parameters (e.g. geometrical parameters, boundary values, etc.)

New Features and Improvements - V2009.0

CFD-ACE-SOVLER - V2009.0.1

CFD-ACE-SOLVER V2009.0.1 includes all the bug fixes and improvements made between June 2008 and October 2008. This is the first Release version of the 2009.0 software. The following sections summarize significant new features and improvements developed since the 2008.2 release.

Fast Time Stepping (FTS) Enhancements

FTS, first released in V2008.2 and based on Fractional Step Method and PISO, offers speedup in convergence within each time-step. The number of iterations needed to convergence is typically reduced by factor of five by using FTS compared to STS. FTS is designed for strongly transient simulations, i.e. those that require small physical time-step compared to the integral time scale. Examples of such simulations are eddy resolving turbulent simulations (LES and DES), free-surface simulations based on volume of fluid (VOF), and time-accurate flows involving Lagrangian particle tracking.

A practical way to determine whether it is more efficient to use FTS or STS to simulate a certain physics problem is to compare the time-step required for numerical stability and the required physical time-step, i.e. the time scale to capture certain physical phenomena. Users ought to use FTS only if the physical time scale is smaller than the stability time-step, otherwise STS should be used due to its enhanced numerical stability.

In V2008.2, FTS was available only for flow, turbulence and heat transfer. In the current release, V2009.0, the scope of FTS has been extended to include VOF, porous media, chemistry, spray, stress, and grid motion. Other physics may be activated relatively easily within the FTS transient framework based upon user's requests.

In addition to the inclusion of more physics, a new feature has been added in the FTS scheme, namely CFL-Based Auto-Timestepping. Using this feature, users do not have to determine a proper time-step size for the simulation, which usually is not a trivial task. The time-step is determined dynamically as function of local flow conditions based on CFL stability criterion. The recommended range of CFL number is 1 &endash; 3. A lower CFL value gives more stable time-stepping, whereas a higher value gives larger time step size, therefore shorter time to a (statistical) convergence. This guide line is also useful to determine whether to retain FTS or switch to STS. If the physical time scale that needs to be resolved is at least an order of magnitude larger than the stability time-step given by the CFL criterion, then it is probably more efficient to use STS.

Force and Moment Coefficients on Specified Surfaces

This feature controls the output of the forces and moments calculated on walls. It provides users more control over the information to be output by outputting for selected walls or groups of walls. As an example of the use of this feature in the automotive application, for instance, users have the freedom to select only car walls and exclude the floor wall from the total force and momentum. More specific, users can group wall surfaces that form doors, underhood, or side mirrors into different groups to assess forces exerted locally on each of these components. This feature, hence, provides greater flexibility and granularity in accessing the desired information.

The feature outputs the actual dimensional values of forces, moments, as well as their corresponding coefficients. All reference values can be specified by the user or calculated by the code if the user does not specify them, with the exception of the reference length and the reference area which have to be specified by the user. The reference values for velocity, area, and density are required to compute the coefficients for forces, whereas the reference length is

6

Intro-New Features

needed for the coefficients for moments. The reference values for density and velocity are taken as the average at the inlet boundaries if they are not specified by the user, and assumed to be the free-stream values. The output is written to a separate force-moment summary file having the extension FMSUM.

This enhanced output functionality has the following features

outputs forces and moments exerted on walls

outputs force and moment coefficients for walls

allows user to specify position where moments are taken about

allows users to output a summary for every wall, specific walls, or a group of walls

applicable for serial and parallel simulations

Forces and Moments for Momentum Resistance

This feature is provided for the evaluation of aerodynamic forces and moments acting on Momentum Resistance surfaces. The Momentum Resistance model is typically used for specific parts within an application such as car radiators and components made of porous media. Using this tool, user can estimate up to grid resolution precision the drag and torque caused by the presence of these parts in the flow way. This functionality can be activated by checking the ’r;Include Momentum Resistance’ box along with the ’r;Force and Moment Summary’ box in the Out section of the GUI.

The transfinite interpolation technique (TFI) is employed by the Solver to create surface meshes on the momentum resistance. This provides the area and the coordinates of each face of the surface mesh. The pressure at each face of the surface mesh is determined by the background flow field pressure value based on the proximity of the surface face centers to the background cell centers. Having collected the required data, the pressure forces and moments for each momentum resistance are then computed in a straightforward manner.

This tool contains the following features:

multi-components (multiple momentum-resistances)

normal cross section of momentum resistance can have arbitrary polygon shape

serial and parallel simulations

For an optimum result, the size of background cells should not be too large compared to the size of the momentum resistance. Typically, the normal cross section of the momentum resistance should be big enough to cover at least 100 background cell faces.

V2F Model Enhancement

The k-eps-v2-f model robustness has been enhanced by improving the convergence of the f-equation. This helps the overall convergence character of the model, especially when applied to complex turbulent flows. Users have to be aware, however, that k-eps-v2-f model, although offering more accurate and realistic solutions for flows more challenging to other turbulent models, is known to be less ’r;code-friendly’ than other RANS models based on transport equations. This means that users should expect slower convergence using this model compared to other Boussinesq type models, with the trade off of capturing specific feature the other models fail to capture. A well known example for this is turbulent separating flow in an asymmetric plane diffuser (ref. 1 and 2).

1. Obi, S., Aoki, K., and Masuda, S., 1993, ”r;Experimental and Computational Study of Turbulent Separating Flow in an Asymmetric Plane Diffuser”, Proc. 9th Symposium on Turbulent Shear Flows, pp. 305-312.

2. Buise, C. U., and Eaton, J. K., 1997,”Experimental Investigation of Flow Through an Asymmetric Plane Diffuser”, Report No. TSD-107. Thermo-sciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, USA.

7


Two-Phase Model Improvement

A number of algorithms can be used for the solution of the phase-fraction equation. In the One Equation Model, the continuity equation for the dispersed phase is solved to obtain alpha and the constraint equation, i.e. the sum of the phase fractions must be unity, is used to obtain the solution for the primary phase. This method does not guarantee boundedness for both phase fractions. In the Two Equation Model, the continuity equations of both phases are solved separately. The solution is then renormalized to ensure a unity sum. The principle disadvantages of the Two Equation model are that this approach does not guarantee continuity of the individual phases until overall convergence, due to the renormalization, and the increased computational effort required for solution of an additional scalar equation.

In the new release, the approach of composite solution of and was followed. This approach derives an equation for the phase fraction by taking the difference of the density-weighted continuity equations. The continuity equation for each phase is solved with the

constraint .

Ballard Power Systems channel geometry utilized for algorithm evaluation, shaded by a

typical pressure distribution

The selected channel (figure above), is a Ballard Power Systems design. The convergence of steady state two-phase flow simulations was evaluated for the old and new formulations. The convergence histories, shown in the figures below, clearly demonstrate the advantage of the new algorithm for this particular application.

Improved Convergence with new formulation

Two Phase Porous Media

The two-fluid model has been extended to porous media. The model presents an attractive alternative to VOF for simulations where accurate interface information is not critical. The fluid-solid momentum transfer is given by a generalized form of Darcy’s Law. The effect of surface-tension will be included in subsequent releases.

8

Intro-New Features

9

Two-phase Pressure drop in non-porous (left) and porous (right) channels

CFD-ACE-GUI - V2009.0.1

CFD-ACE-GUI V2009.0.1 includes all the bug fixes and improvements made between June 2008 and October 2008. This is the first Release version of the 2009.0 software. The following are the new features in this release.

ESI Mobile Support

ESI Mobile is an exiting new application for giving client-server access to engineering simulations. By extending your control over your simulations, you will find increased insight into your models whether you’re at your desk, in the conference room down the hall, in an airport, or at a

customer’s site on the other side of the planet!

This version extends the remarkable capabilities of ESI Group’s flagship Computational Fluid Dynamics (CFD) applications into the iPhone & iPod Touch world. This iPhone App is a free extension of CFD-VIEW and CFD-GUI, the ESI applications for viewing CFD simulation results and CFD model setup.

ESI Mobile uses the latest features and thiPhone interface to provide a very intuitway of monitoring and controlling your simulation remotely.

e ive

iew r

le makes use of the latest ing

With ESI Mobile you can start and stop the solver, change simulation parameters, vsolver output, watch residual and monitopoint plots in real time on your iPhone.

ESI Mobiindustry-standard protocols for securyour information, and for making setup assimple as launching the App. Visualization and Simulations servers are automatically located on your LAN, or can be found at remote locations with ease.

Using Help

This help system is arranged in two volumes:

Volume I - User Manual describes the CFD-ACE+ operations and features of the CFD-ACE-Solver which are module independent:

Volume II - Modules contains a section for each of the CFD-ACE+ modules that appear in the Problem Type (PT) Panel:

User Manual Overview

Database Manager

Arbitrary Interface Boundary Conditions

Thin Wall Boundary Conditions

Cyclic Boundary Conditions

Periodic Boundary Conditions

Fan Model

Momentum Resistance

Porous Media

Rotating Systems

Parallel Processing

User Subroutines

Numerical Methods

Mixing Plane

Filament Model

Electrokinetics

Ionization

Electroplating

Dielectrophoresis (DEP)

Solidification

Fuel Cell Modeling

Biochemistry

Appendix A - CFD-ACE+ Files

Appendix B - DTF Utility

Appendix C - GUI Scripting

Flow

Heat Transfer

Turbulence

Chemistry

User Scalar

Radiation

Cavitation

Grid Deformation

Stress

Electric

Magnetic

Spray

Macro Particle

Free Surface (VOF)

Plasma

Two-Fluid

Kinetic

Semi Device

We recommend that you first read the User Manual Overview to learn the basics of how the CFD-ACE+ application works. Then review the remaining information in the User Manual and Modules that apply to your application of interest for details on using each module or feature.

It is also worthwhile to review the Introduction, Applications, and Features sections of each module to determine if they can help you to model your systems.

10

Getting Started

How to execute the software:

To execute the graphical software (once the environment and path has been set according to the installation instructions that can be found on the CFD Portal) from the command line, enter one of the following commands in a DOS window on Windows Systems or in a shell on Linux/UNIX systems:

CFD-GEOM

CFD-CADA

CFD-VisCART

CFD-ACE-GUI

CFD-FASTRAN-GUI

CFD-TOPO-GUI

CFD-VIEW

SimManager

The appropriate solver can be executed from CFD-ACE-GUI, CFD-FASTRAN-GUI, CFD-TOPO-GUI, or SimManger. They can also be submitted from the command line using:

CFD-ACE-SOLVER –dtf model.DTF

CFD-FASTRAN-SOLVER –dtf model.DTF

CFD-TOPO-SOLVER –dtf model.DTF

If multiple versions of the software have been correctly installed, then the old version can be executed using: CFD-GEOM –runver 2006 (which will run version 2006 of GEOM).

Note your license file will dictate which applications you can execute.

How to add shortcuts to the Start Menu:

Windows users that installed via CD will have short cuts under Start -> Programs -> ESI-Software. If your software was received via ftp or the CFD portal, then you can create your own short cuts. To do so:

1. Create an ESI_Software folder typically under C:\Documents and Settings\All Users\Start Menu\Programs

2. Copy the desired icons from the latest UTILS_20xx.x\icons directory in the ESI_Software folder

3. In Windows Explorer, right click on the icon and select: Create Shortcut

4. Right click on the just created shortcut and select: Properties

5. Change the target to the desired application in the UTILS_20xx.x\bin directory (for instance: CFD-VIEW.exe)

6. Change the Start in directory to your desired starting location

7. Select the Change Icon button and browse back to the originally icon in the UTILS_20xx.x\icons directory and select the appropriate icon.

8. Delete the icon that is setting Start Menu\Programs directory

11


12

9. Repeat as needed

Note the target string can contain at the end the –runver option (ie. –runver 2006) so that a specific version of the software can be executed. If this option is not specified, by default the latest version found will be executed.

Where to request a license file:

The following table gives the email address where to request a license key:

Country/Region Contact North America [email protected] South America [email protected] France Rest of Europe [email protected] Japan [email protected] Korea [email protected] China [email protected] India [email protected] Rest of Asia [email protected] Australia/New Zealand [email protected] Africa [email protected]

Or contact your local ESI Sales Representative.

How to Report Problems

If you experience problems while using CFD-ACE-GUI/SOLVER, you can report your problem by:

E-mail: [email protected]

Telephone: 256-713-4750 (United States country code is 01)

When reporting a problem it is important to have the following information:

CFD-ACE-GUI version number

CFD-ACE-Solver version number

Modules you were using

Type of problem you were trying to solve

Any error messages you may have received in the modelname.out file or screen

To find the CFD-ACE-GUI version number or the DTF version number:

1. Open the CFD-ACE+ application.

2. Click on the Help menu.

3. Select the About CFD-ACE-GUI option.

4. Make a note of the version number.

-OR-

1. Open a command prompt.

2. On the command line, enter CFD-ACE-GUI -v and press Enter. (Note that the command is CFD-ACE-GUI(space)-v and the command is case sensitive in most environments) A file is created (CFD-ACE-GUI.version) which contains the build date and version information. The file will be created in the current working directory.

To find the CFD-ACE-Solver version number:

1. Note the version number at the top of the CFD-ACE-GUI modelname.out file.

-OR-

1. Open a command prompt.

2. On the command line, enter CFD-ACE-SOLVER -v and press Enter. (Note that the command is CFD-ACE-SOLVER(space)-v and the command is case sensitive in most environments) A file is created (CFD-ACE-SOLVER.version) which contains the build date and version information. The file will be created in the current working directory.

13


Flow Module

Flow Module Introduction to Flow Module

The Flow module is the heart of CFD-ACE+ and is used in most simulations. Activating the Flow Module implies the solution of the velocity field by solving for the x, y, and z momentum equations, and the pressure field by solving the pressure correction equation. You can use the Flow module with one or more of the CFD-ACE+ modules to provide a multi-physics based solution to an engineering problem (such as couple flow with heat transfer, mixing, finite-element stress solution, etc.). The Flow module includes the following sections:

Applications

Features and Limitations

Theory

Implementation

Frequently Asked Questions

Examples

References

Flow Module Applications of the Flow Module

The Flow Module allows CFD-ACE+ to simulate almost any fluid (gas or liquid) flow problem. Both internal and external flows can be simulated to obtain velocity and pressure fields. Below are some examples of applications that use the Flow Module exclusively as well as a list of other modules that can be used together with this module to produce a multi-physics simulation. It is assumed that the flow is laminar unless the Turbulence Module is activated.

Flow Visualization

CFD-ACE+ flow solutions can be used to provide detailed information about the flow field. For example, vector plots can be used to depict the magnitude and direction of the flow velocity. Streamline traces can also be produced to show how the flow progresses through the solution domain.

Pressure Field Calculations

The Flow Module is often used to determine the pressure field within a given geometry. Using CFD-ACE+ to predict the pressure drop through a device can help determine the amount of power needed to drive the flow. For external flow applications, the pressure field can be used to obtain pressure forces acting upon the body.

Mass Flow Calculations

The Flow Module solves the velocity and pressure equations, and hence can be used to determine the mass flow characteristics of an internal flow system. CFD-ACE+ can determine the mass flow rate through a system for a given differential pressure. Mass flow calculations are also useful for determining flow splits when the flow is bifurcated.

Multi-Physics Applications

14

Flow Module

You can use the Flow Module with many of the other CFD-ACE+ modules to perform multi-physics analyses. The more commonly added modules are listed below. Examples of these types of applications are given in each module’s section.

Turbulence (Flow is required)

Heat Transfer (with or without radiation)

Chemistry (Flow is required)

With or without gas-phase and surface reactions

Biochemistry

User Scalar

Spray (Flow is required)

Free Surfaces (Flow is required)

Two Fluid (Flow is required)

Cavitation (Flow is required)

Grid Deformation

Finite Element Stress

Plasma (Flow is required)

Kinetic

Electric and Magnetic Module (Electrophysics)

Flow Module Features and Limitations of the Flow Module

Features

The Flow Module has many inherent features that may or may not be activated for any given simulation.

Non-Newtonian Viscosity Options

The Flow Module can model non-Newtonian flows through the use of power law and Carreau law viscosity property options. (See Volume Conditions for details on activating non-Newtonian viscosity properties).

Swirl Model

A swirl model provides a solution for tangential velocity (W) in 2D-axisymmetric geometries. This feature can be used to yield 3D results from a 2D axisymmetric computational grid system, thus saving computational resources.

Slip Wall Boundary Conditions

The default treatment for wall boundary conditions is the no-slip condition for momentum and heat transfer (i.e., all velocity components are set to the wall velocity, usually zero, and the gas temperature is set to the wall temperature). However, at low pressures (on the order of 1 mTorr) the no-slip boundary condition is no longer appropriate. For this reason a slip wall feature is included that allows for velocity slip and temperature jump at the walls. (See Theory-Slip Walls for details on how to activate this feature).

Hemolysis Model

15


Shear stress exerted on the blood may damage or destroy the red blood cells. The phenomena of destruction of red blood cells and subsequent release of hemoglobin is called hemolysis. This occurs commonly when vascular access is made using a vascular device, for example using a needle or a catheter. Hence, it is essential to operate these devices a safe operating mode so that the maximum shear stress is well below the threshold stress for hemolysis. The geometry and orientation of the vascular devices may cause the maximum stresses to occur at the device wall rather than the venous wall. Hence, the classical Poiseuille theory for predicting maximum shear stress cannot be applied to assess the device performance. Secondly, the sub-threshold damage to the cells may accumulate over the time and account for delayed hemolysis. An explicit Lagrangian type particle tracking scheme is developed in CFD-ACE+ as a post processing tool, to compute mass-averaged hemolysis index. See the Hemolysis Model for details on how to activate this feature.

Simple Flow Models

Reduced flow models are provided to take advantage of theoretical assumptions to include more physics in the solution process. Their use is only applicable to a small set of problem types. However, when used they can produce highly accurate results with less computational resources. See Simple Flow Models for the theory behind these models and for details on how to implement this feature.

Limitations

Although the Flow Module can handle compressible flows, the pressure-based method that CFD-ACE+ uses is not ideally suited to higher supersonic flows. CFD-ACE+ has been validated for supersonic flows with Mach numbers on the order of two. For higher Mach number flows, use a density-based solver like CFD-FASTRAN.

Theory

Flow Module Flow Module Theory

The governing equations for the Flow Module represent mathematical statements of the conservation laws of physics for flow.

The mass of a fluid is conserved, i.e. there is no loss or gain of mass in the system.

The time rate of change of momentum equals the sum of the forces on the fluid (Newton’s second law).

You can use these two laws to develop a set of equations (known as the Navier-Stokes equations), which CFD-ACE+ to solves numerically using an iterative method.

Mass Conservation

Momentum Conservation

Navier-Stokes Equations

Mass Conservation[1]

Conservation of mass requires that the time rate of change of mass in a control volume be balanced by the net mass flow into the same control volume (outflow - inflow). This can be expressed as:

(1-1)

16

Flow Module

The first term on the left hand side is the time rate of change of the density (mass per unit volume). The second term describes the net mass flow across the control volume’s boundaries and is called the convective term.

Momentum Conservation[1]

This section describes the mathematical equations used by the Flow Module. See Numerical Methods for details on the methods used to solve these equations.

Newton’s second law states that the time rate of change of the momentum of a fluid element is equal to the sum of the forces on the element. We distinguish two types of forces on the fluid element:

Surface forces

Pressure forces

Viscous forces

Body forces

Gravity force

Centrifugal force

Electromagnetic force

Surface tension force

Momentum resistance

Porous media forces

This section describes the surface forces. The body forces are included as source terms and are discussed in the chapters for gravitational and rotational body forces (see Rotating Systems ). Also see Magnetic Module for information on the body forces produced by that module.

The x-component of the momentum equation is found by setting the rate of change of x-momentum of the fluid particle equal to the total force in the x-direction on the element due to surface stresses plus the rate of increase of x-momentum due to sources:

(1-2)

Similar equations can be written for the y- and z-components of the momentum equation:

(1-3)

(1-4)

In these equations, p is the static pressure and ij is the viscous stress tensor.

Navier-Stokes Equations[1]

The momentum equations, given above, contain as unknowns the viscous stress components ij, therefore a model must be provided to define the viscous stresses.

17


In Newtonian flows, the viscous stresses are proportional to the deformation rates of the fluid element. The nine viscous stress components (of which six are independent for isotropic fluids) can be related to velocity gradients to produce the following shear stress terms:

(1-5)

(1-6)

(1-7)

(1-8)

(1-9)

(1-10)

Substitution of the above shear stress terms into the momentum equations yields the Navier-Stokes equations:

(1-11)

(1-12)

(1-13)

By rearranging these equations and moving the smaller contributions of the viscous stress terms to the momentum source term, we can rewrite the Navier-Stokes equations in a more useful form:

(1-14)

(1-15)

(1-16)

18

Flow Module

For more details on the discretization of these equations and the method used to obtain velocity-pressure coupling please see the Numerical Methods.

Flow Module Implementation

Flow Module Simple Flow Model Theory

There are two simple flow models available for use by the Flow Module:

Second Order Wall Model - used to calculate the shear stress at the wall more accurately. This model can be used on multi-cell thick grid systems.

One Cell Wall Model - intended for fully developed channel flows to be simulated with a single cell thick grid (multiple cells in the stream-wise direction).

Second Order Wall Model

The Second Order Wall model differs from the One Cell Wall model in the way that the coefficients in equation 1-17 are determined. Instead of applying equation 1-17 between two opposite walls, the Second Order Wall model applies the equation only in the cell adjacent to the wall. The relevant boundary conditions to determine the three coefficients in equation 1-17 are:

at the wall

at the cell center

at the cell center

With these boundary conditions the coefficients for equation 1-17 can be calculated. The rest of this model is the same as the One Cell Model.

One Cell Wall Model

Channel flows have specific characteristics. For low values of the Reynolds number, the flow creates a laminar boundary layer near the wall. If the channel aspect ratio is large enough, i.e. the length divided by the height is very large, the flow inside the channel will often be fully developed. The One Cell Wall simple flow model takes advantage of the assumption of fully developed flow to impose analytical models on the flow solution.

The One Cell Wall model uses the assumption that the flow velocity distribution between two walls facing each other (parallel or not) achieves a parabolic profile. This assumption is derived from fully developed channel flows and is true for laminar flow between straight parallel walls. It is approximately true for flows between non-parallel walls. However, if the walls change their shapes smoothly, the assumption may still be valid.

Assuming laminar flow passing between two walls, the boundary conditions are U = U1 at wall 1 and U = U2 at wall 2. Since the velocity distribution is parabolic, we prescribe a parabolic profile for velocity as:

(1-17)

where a, b, and c are constants and y is the local distance normal to the wall. The shear stress therefore can be calculated as:

19


(1-18)

From the boundary conditions, we can determine two of the constants in equation 1-17. The third constant can be determined if we know the value of U at the center which can be obtained by solving the Navier-Stokes equations. CFD-ACE+ solves the Navier-Stokes equations to get the cell volume average value of the velocity. If we use this average velocity to compute the shear stress, we will get a stress that is lower than the actual value. Therefore, we add one more condition to determine the third constant:

(1-19)

where is the average value and obtained from the standard numerical solution process and is the difference between the real value and the numerical solution. The value U can then be determined from the following equation:

(1-20)

where V is the cell volume. Once U is determined, we can compute the shear stress and use the new shear stress as a boundary condition for the Navier-Stokes solution.

Note: For the One Cell Model, the velocity in CFD-VIEW will appear to be zero. This is due to the fact that all the nodes lie on a wall boundary, and the velocity at the wall is zero for no-slip conditions.

Flow Module Slip Wall Theory

Assuming that the Us is the slip velocity, Uw is the wall velocity (for a stationary wall, Uw= 0), the slip boundary condition formulation as adopted in CFD-ACE+ is:

(1-21)

where:

20

Flow Module

and

= accommodation coefficient (user input)

= the mean free path (calculated in the code, as a function of molecular diameters, local pressure, temperature )

n, n0 = constants

= constants

= normal velocity gradient at the wall

For temperature slip, we have:

(1-22)

Model Setup

Flow Module Implementation and Grid Generation

The Implementation section gives details about how to setup a model for simulation using the Flow Module. The Flow Module Implementation section includes:

Model Setup and Solution - Describes the Flow Module related inputs to the CFD-ACE-SOLVER

Post Processing - Provides tips on what to look for in the solution output

21


The following geometric systems are supported by the Flow Module: 3D, 2D Planar, 2D Axisymmetric. All grid cell types are supported (quad, tri, hex, tet, prism, poly).

The general grid generation concerns apply, i.e., ensure that the grid density is sufficient to resolve solution gradients, minimize skewness in the grid system, and locate computational boundaries in areas where boundary values are well known.

For pure flow problems, most gradients will be located near walls and free shear layers. Also, be aware of streamwise flow gradients, which can be encountered in developing flows and compressible flows with shocks. It is important to pack the grid in any location where solution gradients are expected (e.g. the bend of a pipe).

Flow Module Problem Type

Click the Problem Type [PT] tab to see the Problem Type Panel. See Control Panel-Problem Type for details.

Select Flow to activate the Flow Module. The Flow Module is required for most simulations and can also be coupled with virtually all other modules.

Model Options

Flow Module Model Options - Shared Tab

Under the Shared Tab, a title can be given to the simulation, the gravity vector can be specified, and the frame of reference in which to perform the calculations can be specified.

When gravity is selected, the vector along with the reference density must be specified. For more information on how to set the reference density, please refer to the FAQ section of the Flow Module.

When Rotation Reference is chosen, two options are available: VC Based and Global (Absolute). The VC Based option allows for multiple reference frames to be used, i.e. each volume in the model could have it's own reference frame. The Global (Absolute) option perform all calculations in the rotating frame of reference. For more information on setting the Rotation Reference, please refer to the Rotating Systems chapter.

Also, for 2D cases there is an option for running the case axi-symmetric (about the X-axis). The grid must be in the positive y direction.

If you want to run a simulation using Chimera, simply activate the Chimera option then make the appropriate Chimera settings in the VC and BC tabs. For more information on the Chimera Grid option, please check the Chimera Grid Methodology chapter.

Flow Module Model Options - Flow Tab

Reference Pressure

The Flow Module in CFD-ACE+ enables you to specify a reference pressure (Pref). The value specified for Pref will be added to any pressure inputs (i.e., boundary conditions and initial conditions). The reference pressure is also subtracted from the pressure field for graphical output purposes. This feature enables you to perform your simulation with either gage or absolute pressures.

The default reference pressure is 100000 N/m2 (~1 atmosphere). To work with absolute pressures set the reference pressure to 0 N/m2

22

Flow Module

.

Model Options in Flow Module Settings Mode

Hemolysis

For blood flow simulations, activate the Hemolysis model. The hemoglobin released by the flow induced shear forces is a function of the magnitude of shear stress and the exposure time of red cells to the shear field. In CFD-ACE+, an empirical model proposed by Giersiepen et. al has been used as the default model. In this model, the hemoglobin released by red cells is expressed as

(1-23)

where:

= shear stress in N/m2 t = exposure time in seconds A = 3.62x10-5 B = 2.416 C = 0.785

The model is valid even for exposure times below 7 ms. You also have the option of using other empirical models by changing the constant and the exponents in equation 1-23 by way of CFD-ACE+.

Swirl

A swirl model provides a solution for tangential velocity (W) in 2D-axisymmetric geometries. This feature can be used to yield 3D results from a 2D axisymmetric computational grid system, thus saving computational resources.

Flow Module Model Options - Advanced Tab

The Advanced tab allows for the specification of models which are useful for specific types of simulations.

Simple Flow Models

Selecting the Simple Flow Model checkbox allows activation of these models (see Simple Flow Models). There are three options available:

23


1. High Order Wall Local - enables you to assign a unique simple model to each of the walls in the simulation (see Boundary Conditions-Walls for more details).

2. One Cell Wall Global - applies the one cell wall reduced model to all walls in the simulation. The one cell wall model should only be used for low Reynolds number flows and only on single cell thick grid systems.

3. 2nd Order Wall Global - applies the second order wall simple model to all walls in the simulation.

Slip Walls

For low pressure flow simulations (on the order of 1 mTorr) the no-slip boundary condition for velocity and temperature is no longer appropriate. For slip flow regime, the gas viscosity usually needs to be modified based on Knudsen number as follows [6]:

(1-24)

where:

= gas viscosity a and b = constants

Kn = Knudsen number

Flow Module Volume Conditions

Click the Volume Conditions [VC] tab to see the Volume Conditions Panel. See Control Panel-Volume Conditions for details. Before any volume condition information can be assigned, one or more volume condition entities must be made active by picking valid entities from either the Viewer Window or the VC Explorer.

General flow sources can be specified by changing the volume condition setting mode to Flow. Mass sources and momentum sources can be added to the system. There are several types of sources that can be applied: Fixed Source (Volumetric), Fixed Source (Total), Fixed Value, General Source (Volumetric), General Source (Total), and through a user subroutine (USOURCE). See Source Term Linearization for more details on setting general sources and see Momentum Resistance or Source and Fan Model for details on other types of flow sources. For more information on the different types of sources available, please refer to Direct Specification of Source Terms in the Numerical Methods chapter.

With the volume condition setting mode set to Properties select any volume conditions and ensure that the volume condition type is set to Fluid. Only volume conditions that are of type Fluid need to have flow properties specified (since there is no flow in solid or blocked regions there are no fluid properties for those regions.)

There are two volume condition properties required by the Flow Module; density and viscosity. Both density and viscosity can be evaluated using several methods. The methods used to evaluate these properties and the required inputs are given below. To jump to a particular property evaluation method, please select one from the list.

Density: Viscosity: Constant Constant (Kinematic) Mix Polynomial in T Ideal Gas Law Constant (Dynamic) Mix Polynomial in T (Liq) Polynomial in T Sutherland's Law Power Law Piecewise Linear in T Polynomial in T Carreau Law

24

Flow Module

Mix Piecewise Linear in T

Piecewise Linear in T Power Law (Blood)

Mix Polynomial in T Mix Kinetic Theory Casson Model (Blood) Cavitation Model Mix Sutherland's Law Walburn and Schneck (Blood) User Subroutine (UDENS)


Fluid Properties

Density

Constant

The constant options allows for the specification of the density. This option can be used when density variations in the fluid are minimal. For liquids, the density can be specified as constant since they are nearly incompressible.

Required Module (s): Flow Required Input (s): Density in kg/m3

Ideal Gas Law

When compressible effects are not negligible, the Ideal Gas Law should be used. The Ideal Gas Law is given by:

where pref is the reference pressure, p is the calculated static pressure, MW is the species or mixture molecular weight, R is the universal gas constant, and T is the temperature.

Required Module (s): Flow Required Input (s): Molecular Weight in kg/kmol

Polynomial in T

This option will calculate the density as a function of temperature using a polynomial.

Required Module (s): Flow Required Input (s): Polynomial Coefficients

Piecewise Linear In T

This option is available when the Heat Transfer Module is activated. The temperature and the corresponding density at that temperature must be input, which the CFD-ACE-SOLVER will take and use to interpolate between values to set the density. The interpolation is done as follows:

Required Module (s): Flow

25


Required Input (s): Data pairs of Temperature and Density

Mix Polynomial in T

The density of the mixture is evaluated as

where

is the density of the species i as a function of temperature.

Required Module (s): Flow, Chemistry Required Input (s): Polynomial Coefficients for each species used in the model. The values need to be entered in the Database Manager under the Species Physical Tab.


The Mix Piecewise Linear in T option calculate the density of each species in the same manner as the Piecewise Linear in T option. The mixture density is then calculated as:

Required Module (s): Flow, Chemistry Required Input (s): Data pairs of Temperature and Density for each species used in the model. The values need to be entered in the Database Manager under the Species Physical Tab.

Cavitation Model

Using the Cavitation model, the density calculated is a mixture density, i.e. a mixture of vapor and liquid. The mixture density (r) is a function of the vapor mass fraction (f), which is computed by solving a transport equation simultaneously with the mass and momentum conservation equations. The mixture density is calculated using the following relationship:

where v is the vapor density and l is the liquid density. Note that, if the Cavitation module is activated all fluid volumes must use the Cavitation model for evaluation of the density. For more information on this model, please refer to the Cavitation Module chapter.

Required Module (s): Flow, Cavitation Required Input (s): Absolute Saturation Pressure, Liquid Phase Density, Vapor Phase Density.

26

Flow Module

User Subroutine (UDENS)

This option is available for implementing a user defined evaluation for density if the option is not available through CFD-ACE-GUI. The user subroutines required for setting the density are UDENS and UDRHODP. UDRHODP is required to include compressibility of the flui d. For an incompressible fluid, set DRHO_DP to a very small number (~ 1E-20). For more information on user defined volume condition (property) routines, please refer to the volume condition routine section of the User Subroutines chapter.

Viscosity

Constant (Kinematic)

The kinematic viscosity is given as follows:

where is the dynamic viscosity and is the density of the fluid.

Required Module (s): Flow Required Input (s): Kinematic Viscosity in m2/s

Constant (Dynamic)

The dynamic viscosity is given as follows:

where is the density of the fluid and is the kinematic viscosity.

Required Module (s): Flow Required Input (s): Dynamic Viscosity in kg/m-s

Sutherland's Law

Sutherland's Law is given as follows:

where A and B are constants. The default value of A is 1.4605E-06 kg/m-s-K1/2 and B is 112K. These values are for air at moderate temperature and pressures.

Required Module (s): Flow Required Input (s): Coefficients A and B

Polynomial in T

The Polynomial in T option is given as follows:

27


where C0, C1, C2, C3, C4, and C5 are coefficients.

Required Module (s): Flow Required Input (s): Coefficients C0, C1, C2, C3, C4, and C5

Piecewise Linear in T

The Piecewise Linear in T option will linearly interpolate between the specified viscosity and temperature data.

Required Module (s): Flow Required Input (s): Number of data pairs, Temperature, Dynamic Viscosity (kg/m-s)

Mix Kinetic Theory[1]

The Mix Kinetic Theory option will use the kinetic theory of gases to calculate the viscosity of the gas or mixture of gases. For a pure monatomic gas, the viscosity is defined as

where

i = dynamic viscosity of species i

MWi = molecular weight of species i

T = temperature in Kelvin

i = characteristic diameter of the molecule in Angstroms

= collision integral

The collision integral, , is given by

where T* is the dimensionless temperature and is given by

where is the characteristic energy, is Boltzmann's constant, and T is the temperature. To calculate the mixture viscosity using kinetic theory, the following equation is used:

where:

xi ,xj = mass fraction of species i and species j

= viscosity of species i

28

Flow Module

= dimensionless quantity

and i,j is given by:

Required Module (s): Flow, Chemistry Required Input (s): Molecular Weight of each species, Characteristic Energy, and Collision Diameter. These quantities must be input in the Database Manager for each species.

Mix Sutherland's Law

The Mix Sutherland's Law option is applicable when multiple species are present in a system. The viscosity for each species is calculated using Sutherland's Law, which is shown above. The mixture viscosity is then calculated using mix kinetic theory of gases.

Required Module (s): Flow, Chemistry Required Input (s): Molecular Weight of each species, Characteristic Energy, Collision Diameter, and the A and B coefficients for Sutherland's Law. These quantities must be input in the Database Manager for each species.


This method will use the temperature and viscosity data pairs to linearly interpolate the viscosity for each species. Once all the species viscosities have been determined, the mixture viscosity is calculated using mix kinetic theory.

Required Module (s): Flow, Chemistry Required Input (s): Temperature and Viscosity data pairs. These quantities must be input in the Database Manager for each species.

Mix Polynomial in T

This method will use a polynomial, just like in the Polynomial in T method above, to calculate the viscosity of each species. Once all the species viscosities have been determined, the mixture viscosity is calculated using mix kinetic theory.

Required Module (s): Flow, Chemistry Required Input (s): Coefficients C0, C1, C2, C3, C4, and C5 for each species

Mix Polynomial in T (Liq)

29


This option will use a polynomial to calculate the viscosity of each species. Once all the species viscosities have been determined, the mixture viscosity is then calculated using the following formula:

where

xi = mass fraction of species i and species j

= viscosity of species i calculated using a Polynomial in T

Required Module (s): Flow, Chemistry (Liquid) Required Input (s): Coefficients C0, C1, C2, C3, C4, and C5 for each species

Power Law[1]

This option will use a non-newtonian Power Law model to calculate the viscosity of the fluid. The Power Law model is :

where

= the zero shear rate viscosity

K, a1, a2, a3, a4, B

= constants characterizing the fluid

n = the power law index

= temperature

= the cutoff shear rate

= the local calculated shear rate

For a temperature dependent viscosity, a1 or a2 need to be non zero values. If a1, a2, a3, a4, 0, and are set to zero, then the simplest for of the Power Law model is recovered, which is the two-parameter power law given by

[7]

The power law index will determine the classification that the fluid falls in:

n = 1 (the fluid is Newtonian)

n > 1 (a shear thickening fluid (dilatant fluid))

n < 1 (a shear thinning fluid (pseudo-plastic))

Required Module (s): Flow, Chemistry (Liquid)

30

Flow Module

Required Input (s): Mu_0, N, D0, K, A1, A2, A3, A4

Carreau Law[1]

This option will use the Carreau Law model to calculate the viscosity of the fluid. The Carreau Law model is:

[8]

where

= the zero shear rate viscosity

= the infinite shear rate viscosity

n = the power law index

= temperature

a = constant


= the second invariant of the strain rate tensor

If a is two, then the Bird-Carreau model is recovered.

Required Module (s): Flow (Fluid Subtype is Liquid) Required Input (s): Mu_0, Mu_inf, N, K, A

Power Law (Blood)[3]

This model is available when solving for Flow and the subtype of the fluid is liquid. The model is:

where

and


= the consistency constant

= 0.035 (default) {the limiting (Newtonian) viscosity}

= 0.25 (Default)

a = 50 (Default)

b = 3 (Default)

c = 50 (Default)

31


d = 4 (Default)

n 0.45 (Default)

= 1.0 (Default)

This model expects all the inputs in CGS units, since parameters in literature are available in these units). This model has been established for shear rates varying from 0.1 s-1 to 1000 s-1.

Required Module (s): Flow (Fluid Subtype is Liquid) Required Input (s): Mu_inf, Delta Mu, Ninf, Delta N, A, B, C, D

Casson Model[2]

This model is available when solving for Flow and the fluid subtype is liquid. The model is:

where

and


y = the yield stress in shear given by y = (0.0625Hct)3

Hct = the blood hematocrit and should be specified as a fraction between 0 and 1

= a constant

0 = the viscosity of the plasma

The Casson model is normally used for low shear rates (< 10 s-1) and Hct < 40%. The input values for this model should be in CGS units.

Required Module (s): Flow (Fluid Subtype is Liquid) Required Input (s): Muinf, Ninf

Walburn and Schneck[4]

This model is available when solving for Flow and the fluid subtype is liquid. The model is:

where


y = the yield stress in shear given by y = (0.0625Hct)3 Hct = the blood hematocrit and should be specified as a

32

Flow Module

percentage

a1 = 0.00797 (Default)

a2 = 0.0608 (Default)

a3 = 364.625 (Default)

a4 = 0.00499

The constant a3 represents the effect of TPMA (Total Protein Minus Albumin) in the blood and corresponds to a TPMA of 2.6g/100mL. The Walburn-Schneck model has been developed for a TPMA range of 1.5-3.8 g/100mL. If necessary, the constant a3 can be linearly scaled to model blood with a TPMA different from 2.6g/100mL. The Walburn-Schneck model has been validated for a Hct range of 35-50% (common physiological range) and a shear rate ranging from 30-240s-1.

Required Module (s): Flow (Fluid Subtype is Liquid) Required Input (s): A1, A2, A3, A4, and Hematocrit

Boundary Conditions

Flow Module Boundary Conditions

Click the Boundary Conditions [BC] tab to see the Boundary Conditions Panel. See Control Panel-Boundary Conditions for details. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or the BC Explorer.

The Flow Module is fully supported by the Cyclic, Thin Wall, and Arbitrary Interface boundary conditions. (See Cyclic Boundary Conditions, Thin-Wall Boundary Conditions, or Arbitrary Interface Boundary Conditions for details on these types of boundary conditions and instructions for how to implement them.)

All of the general boundary conditions for the Flow Module are located under the Flow tab and can be reached when the boundary condition setting mode is set to General. Each boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). See Control Panel-Boundary Condition Type for details on setting boundary condition types. This section describes the implementation of each type with respect to the Flow Module. The Boundary Conditions section includes:

Inlets

Outlets

Walls

Rotating Walls

Symmetry

Interfaces

Thin Walls

Cyclic

Flow Module Boundary Conditions - Inlets

For any inlet boundary condition the Flow Module ultimately needs to know how to set the velocity, density and temperature for each cell face on the boundary condition patch. There are various ways to specify this information and there are four methods (known as subtypes) available in CFD-ACE+:

33


Fixed Velocity

Fixed Mass Flow Rate

Fixed Total Pressure

Fixed Pressure

Inlet Sub Type

Fixed Velocity

This inlet subtype allows you to set the velocity, pressure (used only to calculate inlet density), and temperature for each boundary face on the inlet to a fixed value (this effectively fixes the mass flow rate). The velocity vector is specified directly and the code calculates the density using the specified values of pressure (P) and temperature (T) and the selected density method (specified in the volume condition settings). For constant density flows, the pressure value is not used.

Fixed Mass Flow Rate

This inlet subtype allows you to specify the velocity direction, pressure (used only to calculate inlet density), temperature, and the total mass flow rate to be applied over the entire boundary patch. The velocity direction is specified directly and the code calculates the density using the specified values of pressure (P) and temperature (T) and the selected density method (specified in the volume condition settings). For constant density flows pressure is not used. The velocity magnitude of each boundary face is determined by scaling the specified magnitude (determined from the specified direction vector) to ensure that the desired mass flow rate is obtained. The same scale factor is applied for all boundary faces on the inlet and is calculated as:

(1-25)

where:

= the specified total mass flow rate

= the vector direction (Nx, Ny, Nz)

The local velocity magnitude can then be determined by applying the same scale factor to all boundary faces:

(1-26)

Fixed Total Pressure[1]

This inlet subtype allows you to fix the total pressure (Po) and total temperature (To) at the boundary patch. For ideal gases, the total temperature and pressure are computed using:

34

Flow Module

(1-27)

(1-28)

where M is the Mach number. For incompressible flows the total pressure is computed using:

(1-29)

Fixed Pressure

For fixed pressure inlets, the velocity is calculated at the cell center and then extrapolated to the boundary face of the inlet. This velocity is used as the inlet velocity, since flow is assumed to be coming into the domain.

Velocity Direction for Inlets

All of the inlet boundary condition subtypes allow for velocity directions to be specified in various ways. There are several ways to specify the velocity directions at inlets:

Cartesian

Normal

Cylindrical

Swirler

The differences in these velocity direction specification modes is given below.

1. Cartesian - Allows you to specify the velocity magnitude in xyz components (U, V, W) for fixed velocity inlets, or the velocity direction components (Nx, Ny, Nz) for fixed mass flow or fixed total pressure inlets.

2. Normal - The code calculates the velocity direction based on the boundary face normal direction. (The face normal always points into the computational domain).

3. Cylindrical - Allows you to specify the velocity direction in axial, radial, and tangential components (Va, Vr, Vt). The axis of the cylindrical coordinate system is always the x-axis.

4. Swirler - Used to simulate swirling flow at an inlet for three-dimensional models. A swirler inlet is a circular or annular inflow region with axial, radial, and tangential velocity components (Va, Vr, Vt). The axis of the swirler is defined by a specified vector (X1, Y1, Z1) -› (X2, Y2, Z2). Any boundary faces that lie within a specified radius from the axis (Ri < r < Ro) will have the swirler condition applied.

The table summarizes the above information by listing the available inlet boundary condition subtypes (with different velocity direction specification options). The table also shows the variables required for each subtype.

Sub Type Required Variables

35


Fixed Velocity (Cartesian) P, T, U, V, [W]3D,2Ds, [Omega]2Ds

Fixed Velocity (Normal) P, T, Vn

Fixed Velocity (Cylindrical) P, T, Va, Vr, [Vt]3D,2Ds

Fixed Velocity (Swirler)3D P, T, Va, Vr, Vt, Ri, Ro, X1, Y1, Z1, X2, Y2, Z2

Fixed Mass Flow Rate (Cartesian)

P, T, Nx, Ny, [Nz]3D,2Ds, [Omega]2Ds, Mdot

Fixed Mass Flow Rate (Normal)

P, T, Mdot

Fixed Mass Flow Rate (Cylindrical)

P, T, Va, Vr, [Vt]3D,2Ds, Mdot

Fixed Mass Flow Rate (Swirler)3D

P, T, Va, Vr, Vt, Ri, Ro, X1, Y1, Z1, X2, Y2, Z2

Fixed Total Pressure (No Direction)

Po, To

Fixed Total Pressure (Normal)

Po, To

Fixed Total Pressure (Cartesian)

Po, To, Nx, Ny, [Nz]3D,2Ds

Fixed Total Pressure (Cylindrical)

Po, To, Va, Vr, [Vt]3D,2Ds

Fixed Pressure P, T

3D Available for 3D simulations. 2Ds Available for axisymmetric 2D swirl simulations.

Flow Module Boundary Conditions - Outlets

For any outlet boundary condition the Flow Module needs to know how to set either the static pressure or the mass flow rate for each cell face on the boundary condition patch. There are various ways to specify this information and for outlet boundary conditions there are four methods (know as subtypes) available:

Fixed Pressure

36

Flow Module

Farfield

Fixed Velocity

Extrapolated

Note that inflow through an outlet can occur anytime during the solution convergence process (even if the final solution indicates all outflow) so it is recommended that you supply a reasonable temperature value. If the final solution shows inflow through an outlet boundary condition, then this indicates that the boundary condition may not have been located in an appropriate place. When this happens an unphysical solution as well as convergence problems may be the result and we recommend that you relocate the outlet boundary condition to an area where there is total outflow if possible.

Also note that for Farfield boundaries, used as intended, inflow through the outlet is perfectly fine. However, for outlet boundaries where only outflow is expected, inflow becomes problematic which indicates that the outlet boundary may not have been located in the appropriate place..

Fixed Pressure

This outlet subtype allows you to specify the static pressure at the outlet location. All other variables (U, V, W, T) will be calculated by the code if the flow at the outlet boundary condition is out of the computational domain.

If the flow happens to be coming into the computational domain at the outlet then the solver treats the boundary condition as an inlet. Hence, you may optionally specify a temperature (T) to be used only in the case that there is inflow through the outlet boundary condition.

Farfield

This outlet subtype can be used if there is a possibility that there is inflow and outflow along the same boundary patch, as might be found at a Farfield boundary of an external flow problem (i.e. a free-stream condition). This subtype is the same as the fixed pressure subtype except that it allows you to specify a velocity, which will be used to calculated the convective momentum flux across the boundary (mdot*backflow_velocity). As such, it has only a small effect on the flow rate, mdot, across the boundary. This velocity will not be used as in the inflow velocity if inflow does occur through an outlet.

Fixed Velocity

This outlet subtype is actually the same as the fixed velocity (Cartesian) inlet subtype, the only difference being that the velocity vector is usually set to be pointing out of the computational domain.

This subtype has been provided as a convenience to specify the mass flow rate at an outlet boundary condition. It is recommended to use this boundary condition only if the mass flow rate at the outlet boundary is known and a fixed total pressure subtype is being used at the inlets. Note that this approach can sometimes produce convergence problems. These problems can sometimes be overcome by running the simulation as transient to a steady state solution.

Extrapolated

This outlet subtype will extrapolate all boundary information from the cell center to the boundary face if the Mach number at the cell center is greater than 1.0. If the Mach number is less than 1.0 then the boundary condition reverts to a fixed pressure subtype and sets the boundary static pressure (P) to that specified. All other comments about the fixed pressure outlet apply to this subtype if the Mach number is less than 1.0.

37


The Extrapolated subtype should only be used when the flow at the outlet is expected to be supersonic.

The table below summarizes the above information by listing the available outlet boundary condition subtypes. The table also shows the required variables and optional variables for each subtype.

Outlet Boundary Condition Subtypes and Variables

Available Subtypes

Required Variables Optional Variables

Fixed Pressure P T

Farfield P, T, U, V, [W]3D,2Ds, [Omega]2Ds

Fixed Velocity P, T, U, V, [W]3D,2Ds, [Omega]2Ds

Extrapolated P, T

3DAvailable for 3D simulations. 2Ds Available for axisymmetric 2D swirl simulations.

Flow Module Boundary Conditions - Walls and Rotating Walls

Walls

The Flow Module requires the specification of velocity for any wall boundary condition, which in most cases is zero. The required flow variables are U, V, and W or Omega (W being present only in 3D and 2D swirl simulations, and Omega present only in 2D swirl simulations.) No-slip boundary conditions are the default settings (i.e, U=V=W=0 at the wall). If the wall is moving then velocity values should be specified.

If you activate the High Order Wall Local simple flow model (see Model Options-Simple Flow Models) then you will have the opportunity to select which reduced flow model to apply to the selected wall boundary condition. The choices are Not Wall Model, One Cell Wall, and Second Order Wall.

Rotating Walls

A rotating wall boundary condition can be used to set a rotational velocity profile on a wall. The required variables for the Flow Module are Cx, Cy, Cz (the xyz location of any point on the axis of rotation), and Wx, Wy, Wz (an omega vector that defines the rotation direction).

If the High Order Wall Local reduced flow model has been activated (see Model Options-Simple Flow Models) then you will have the opportunity to select which reduced flow model to apply to

38

Flow Module

the wall boundary condition. The choices are Not Wall Model, One Cell Wall, and Second Order Wall.

Flow Module Boundary Conditions - Symmetry

The symmetry boundary condition is a zero-gradient condition. Flow is not allowed to cross the symmetry boundary condition. There are no Flow Module related values for symmetry boundary conditions.

Flow Module Boundary Conditions - Interfaces

The interface boundary condition is used to allow two computational regions to communicate information. There are no Flow Module related values for interface boundary conditions.

Interface boundary conditions can be converted to Thin Walls (see Thin-Wall Boundary Conditions). Also see Arbitrary Interface Boundary Conditions for information on other ways for computational domains to communicate.

Flow Module Boundary Conditions - Thin Walls

The Flow Module fully supports the Thin Wall boundary condition. See Thin-Wall Boundary Conditions for instructions on how to setup a Thin Wall boundary condition.

The Flow Module treats a thin wall boundary condition the same as a wall boundary condition (see Walls). Therefore, under the Flow tab, inputs are available for wall velocity specification. This wall velocity will be applied to both sides of the Thin Wall boundary condition.

Flow Module Boundary Conditions - Cyclic/Periodic

Cyclic BC

The Flow Module fully supports the Cyclic boundary condition. See Cyclic Boundary Conditions for instructions on how to setup a Cyclic boundary condition. There are no Flow Module related settings for the Cyclic boundary condition.

Periodic BC

The Flow module fully supports periodic boundary conditions. When periodic boundaries are used, either the pressure drop or mass flow rate must be specified.

Flow Module Initial Conditions

Click the Initial Conditions [IC] tab to see the Initial Conditions Panel. See Control Panel-Initial Conditions for details.

The Initial Conditions can either be specified as constant values or read from a previously run solution file. If constant values are specified then you must provide initial values required by the Flow Module. The values can be found under the Flow tab and the following variables must be set; P, T, U, V, and W or Omega (W being present only in 3D and 2D swirl simulations, and Omega present only in 2D swirl simulations.)

39


If the Heat Transfer Module has been activated, then the Initial Condition for temperature will be located under the Heat tab.

Although the Initial Condition values do not affect the final solution, reasonable values should be specified so that the solution does not have convergence problems at start-up.

For problems with fixed pressure outlet conditions, it is often best to set the initial pressure to the outlet pressure and the initial velocities to some reasonable value. For problems with total pressure inlet conditions, it is often best to set the initial pressure equal to the inlet pressure and the initial velocities to zero.

Flow Module Solver Control Settings

Spatial Differencing Tab

Under the Spatial Differencing tab, you may select the differencing method to be used for the convective terms in the equations. Activating the Flow Module enables you to set parameters for velocity and density calculations. The default method is first order Upwind. See Spatial Differencing Scheme for more information on the different differencing schemes available and Discretization for numerical details of the differencing schemes.

Solver Selection

Under the Solvers tab you may select the linear equation solver to be used for each set of equations. Activation of the Flow Module allows settings for the velocity and pressure correction equations. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 50 sweeps for the velocity equations and 500 sweeps for the pressure correction equation. The default convergence criteria is 0.0001. See Solver Selection for more information on the different linear equation solvers available. See Linear Equation Solvers for numerical details of the linear equation solvers.

Relaxation Parameters

Under the Relaxation tab you may select the amount of under-relaxation to be applied for each of the dependent (solved) and auxiliary variables used for the flow equations. Activating the Flow Module enables you to set the velocity and pressure correction dependent variables, as well as the auxiliary variables; pressure, density, and viscosity. See Under Relaxation Parameters for more information on the mechanics of setting the under relaxation values and Under Relaxation for numerical details of how under-relaxation is applied.

The velocity and pressure correction equations use an inertial under relaxation scheme and the default values are 0.2. Increasing this value applies more under-relaxation and therefore adds stability to the solution at the cost of slower convergence.

The calculations for pressure, density, and viscosity use a linear under-relaxation scheme and the default values are 1.0. Decreasing this value applies more under-relaxation and therefore adds stability to the solution at the cost of slower convergence.

The default values for all of the under relaxation settings will often be sufficient. In some cases, these settings will have to be changed, usually by increasing the amount of under relaxation that is applied. There are no general rules for these settings and only past experience can be a guide.

Variable Limits

Settings for minimum and maximum allowed variable values can be found under the Limits tab. CFD-ACE+ will ensure that the value of any given variable will always remain within these limits by clamping the value. Activating the Flow Module enables you to set limits for the following variables; U, V, W (for 3D or 2D swirl cases), Pressure, Density, and Viscosity. See Variable Limits for more information on how limits are applied.

40

Flow Module

Advanced Settings

Shared

Buffered Output

Higher Accuracy

Flow

There are two settings under the advanced options tab: Cut Diffusion at Inlets and CFL Relaxation.

The Inlet Diffusion option allows you to disable the diffusive link to an inlet boundary. For low pressure transport problems this may be important because it allows you to prevent the diffusive loss of species through an inlet and gives you better control over the amount of each species in the domain since you only have to account for inlet convection.

When using CFL based relaxation, an effective time step is calculated for each computational cell (local time stepping). The size of the cell’s effective time step is calculated by determining the minimum time scale required for convection, diffusion, or chemistry to occur in that cell. This minimum time scale is then multiplied by a user input factor to determine the final effective time step which will be used for that cell.

The default inertial relaxation method can be switched to the CFL based relaxation method by going to SC-->Adv and checking the appropriate check boxes for each module. The relaxation factor defined in SC-->Relax is used as the CFL multiplier.

Rule of Thumb: Inverse value of the usual inertial relaxation factor.

Effect of Value:

5 = Default Value

1 = More stability, Slower convergence

100 = Less stability, Faster convergence

The CFL based relaxation method is not available for all modules.

Flow Module Output Options

The desired output can be specified under the Out (Output) tab. There are two types of output available, Printed and Graphical. Printed output will be written to a text file, whether it be the modelname.out file or another file. Graphical output will be written and stored in the DTF file and available for further post-processing in CFD-VIEW.

Output

For steady state simulations, there are two options for when results will be written to the DTF file. The first is End of Simulation. This option will only write results to the DTF file once the maximum number of iterations has been reached or when the specified convergence criteria is achieved. The Specified Interval option allows for results to be written at certain intervals during the solution process. The file will be uniquely named in the following way: modelname_steady.000025.DTF. A unique file can be created or the results can be written to the same DTF file, modelname.DTF.

For transient simulations, the results will be written out at the specified time step frequency.

41


Printed Output

Under the Print tab, select the printed information to be written to the text based output file (modelname.out). Activating the Flow Module allows output of a Mass Flow summary and a Force summary in addition to the general printed output options. The Diagnostics option will print more information to the modelname.out file, which may be useful if problems are encountered. For more information on the BC Integral Output option, please refer to Appendix A: CFD-ACE+ Files.

The mass flow summary will provide a tabulated list of the integrated mass flow (kg/s) through each flow boundary (inlets, outlets, interfaces (if include interfaces is selected)). The force summary will provide a tabulated list of the pressure and viscous forces (in Newtons) integrated over each solid boundary (walls, fluid/solid interfaces, etc.) as well as the integrated torque (in N-m) over the same boundary conditions about the x, y and z axes.

Graphical Output

Under the Graphics tab, you can select which variables to output to the graphics file (modelname.DTF). These variables will then be available for viewing and analyzing in CFD-VIEW. Activating the Flow Module enables output of the variables listed in the table:

Post Processing Variables

Variable Description Units

U, V, W Velocity Vector m/s

U_absolute, V_absolute, W_absolute

Absolute Velocity Vector m/s

VelocityMagnitude Velocity Magnitude m/s

P Static Pressure N/m2

P_tot Total Pressure N/m2

Vislam Laminar Viscosity kg/m/s

Vorticity Vorticity -

STRAIN_RATE Strain Rate 1/s

RESIDUAL_U X-Direction Velocity Residual kg-m/s2

RESIDUAL_V Y-Direction Velocity Residual kg-m/s2

RESIDUAL_W Z-Direction Velocity Residual kg-m/s2

RESIDUAL_P Pressure Residual kg/s

Flow Module Post Processing

CFD-VIEW can post-process the solutions. When the Flow Module is invoked, the velocity and pressure fields are usually of interest. A complete list of post processing variables available as a result of using the Flow Module is shown in the table. Use CFD-VIEW’s vector plot and stream trace features to view the velocity field. The pressure field can be viewed with surface contours and analyzed through using point and line probes.

42

Flow Module



U, V, W Velocity Vector m/s

U_absolute, V_absolute, W_absolute

Absolute Velocity Vector m/s

VelocityMagnitude Velocity Magnitude m/s

P Static Pressure N/m2

P_tot Total Pressure N/m2

Vislam Laminar Viscosity kg/m/s

Vorticity Vorticity -

STRAIN_RATE Strain Rate 1/s

RESIDUAL_U X-Direction Velocity Residual kg-m/s2

RESIDUAL_V Y-Direction Velocity Residual kg-m/s2

RESIDUAL_W Z-Direction Velocity Residual kg-m/s2

RESIDUAL_P Pressure Residual kg/s

The mass flow summary and force summary written to the output file (modelname.out) are often used to determine quantitative results. The mass flow summary can also be used to judge the convergence of the simulation. Due to the law of conservation of mass, the summation of all mass flow into and out of the computational domain should be zero (unless mass sources or sinks are present). In the simulation a summation of exactly zero is almost impossible, but you should see a summation that is several orders of magnitude below the total mass inflow.

Vorticity is calculated as follows:

The Strain Rate components are shown below. The Strain Rate reported in CFD-ACE+ is the magnitude of all the components.

The following are forces which are written out when the Force Summary is activated under the Printed output option for the flow module.

43


Pressure forces are calculated in the following manner:

where A is the face area, FC is the face normal x-component, y-component, or z-component, and P is the pressure. The shear forces are calculated in the following manner:

where l is the laminar viscosity, A is the face area, Vrel is the relative velocity at the boundary in the x-direction, y-direction, or z-direction, and xN is the distance from the cell center to the face center. The pressure moments are calculated in the following manner:

where FC is the face center location for a given x, y,or z component and P is the pressure force for a given x, y, or z component. The viscous moments are calculated in the following manner:

where FC is the face center location for a given x, y,or z component and Fsh_i is the shear force for a given x, y, or z component.

Flow Module Frequently Asked Questions

Why do I have to specify the pressure at a fixed velocity or fixed mass flow inlet boundary condition?

The Flow Module ultimately needs to set the value of density at the inlet boundary condition. The pressure specified for a fixed velocity or fixed mass flow inlet will only be used to calculate the density at that inlet. Since the inlet pressure is only used to calculate the inlet density, it is not required when the fluid density is constant. Note that the solution results will show the calculated inlet pressure, not the specified inlet pressure. See Fixed Velocity or Fixed Mass Flow Rate for more information.

Why is the velocity zero in CFD-VIEW when I use the one-cell model?

For the One Cell Model, the velocity in CFD-VIEW will appear to be zero. This is due to the fact that all the nodes lie on a wall boundary, and the velocity at the wall is zero for no-slip conditions.

44

Flow Module

What is the reference density? What value should I enter?

Buoyancy-driven flows are those in which density variations cause the fluid motion. Examples include low-pressure mixing of gases and natural convection heat transfer. In CFD-ACE+, you must activate Gravity on the MO/Shared tab if you want to capture buoyancy effects. Gravity is Off by default because hydrostatic pressure variations do not contribute to fluid motion in steady flows, and because the effect of hydrostatic pressure variation on fluid density is usually small (non-existent for incompressible fluids).

Once you’ve activated it, an additional input option appears, asking you to choose how the “Reference Density” is to be calculated. The reference density is explained below and what you need to know to select the write option for your case.

The acceleration due to gravity of a fluid in any given control volume is –ρg. In CFD-ACE+, ρ = ρ0 + ρ', where ρ0 is the reference density, and the gravitational body force is implemented as -ρ'g. Omitting the ρ0g term in the momentum equation produces a pressure field p*, as follows:

In other words, the hydrostatic pressure variation is omitted. This formulation is useful because it simplifies the specification of pressure boundary conditions.

Consider buoyant flow along a heated wall, as shown below in the figure below. The pressure along the open boundary should vary linearly with height, but in order to specify this variation we would have to use a profile boundary condition or a user subroutine. By omitting the ρ0g term, we are able to specify a constant pressure on all 3 open boundaries and set up this type of problem with ease.

The only drawback to this formulation is that you can no longer see hydrostatic pressure variations in CFD-VIEW, only those pressure differences due to the velocity field.

There are two ways to specify the reference density, ‘Automatic’ and ‘User-Specify’. The Automatic option behaves one of two ways, depending on whether the system is open or closed. For open systems such as the example above, reference density is calculated from the initial solution as the average density over all inlet/outlet boundaries. For closed systems, such as a box heated on one side only, ρ0 is the average density over the entire domain.

The Automatic reference density option is not appropriate for every problem involving gravity, only for buoyant flows where the driving forces for fluid motion are density differences. For unstable transient cases where the weight of the fluid causes fluid motion, the ‘User Specify’

45


option should be chosen and the reference density set to zero. In addition, the initial pressure field must include the hydrostatic pressure variation, i.e. must be physically realizable. Most likely you would need a UINIT user subroutine for such cases.

In general, there is no harm in using the Automatic reference density option. However, if there is any doubt, choose the ‘User-Specify’ option and set the reference density to zero, while paying special attention to any pressure boundary conditions, i.e. don’t forget to include hydrostatic variations. Also, be aware that an initial guess of p = 0 everywhere may be very harsh for steady-state cases and can cause convergence problems. Increased velocity relaxation and/or a better initialization of the pressure field can get around such problems.

What settings should I use for natural convection problems with ambient boundaries?

For natural convection problems, it is imperative that the boundary conditions are specified properly. Often, the mistake is in the specification of pressure on an ambient "free" (or outlet) boundary. The common practice is to use a reference pressure that is equal to the ambient pressure and to set the pressure at the free boundaries to zero. This is a correct specification only if the ambient boundaries all have exactly the same elevation. If there is an difference in elevation between the free boundaries then there is a pressure difference between the boundaries which we usually taken to be equal to gh, where h is the difference in elevation.

Incorrect Boundary Conditions

This problem can be seen clearly by simulating a "null" problem - that is, one where the we know the trivial solution to have no temperature difference and no motion. Such a problem is illustrated in Figure 1, where the flow along a vertical flat plate is modeled, but the plate temperature is set to be equal to the ambient temperature.

Figure 1. Test problem conditions

46

Flow Module

If this problem is modeled using the ideal gas law for the fluid density and assign zero pressure to all the free boundaries, the result is clearly incorrect and is shown in Figure 2. Although the temperature field is not shown, it was checked and verified to be a constant of 292K. The resultant velocity field shown in Figure 2 has a maximum down ward velocity of almost 3m/s. This error occurs because the external pressure gradient was neglected when the pressure was specified at the "free" boundaries.

Figure 2. Solution using ideal gas law and zero pressure at boundaries

Correct Boundary Conditions

There are at least three ways to correct the problem definition so that we actually describe the problem we want to solve and obtain the correct solution. They are the follow:

1. Change nothing except the specification of the boundary pressures. This requires a pressure specification that varies with y for the vertical boundary.

2. Assign a reference density to be used in the calculation of the buoyancy source term.

3. Use the Boussinesq approximation.

Boussinesq Approximation

With the Boussinesq approximation, a constant fluid density is used, which is evaluated at the specified reference temperature) and the buoyancy source is calculated as:

S = (T - Tref)*g*Vol

If Tref is set equal to the ambient temperature and equal to 1/Tref, then there will be no externally imposed pressure gradient because there will be no source for cell where T = Tref. What we are doing with this option is subtracting off the hydrostatic pressure variation which does not contribute to fluid motion. If this option is used, the solution will be like Figure 3. We see a

47


downward velocity field predicted, but the velocity magnitude is effectively zero. This would be an acceptable solution to this problem.

Figure 3. Solution with Boussinesq approximation

Use of Reference Density

A second, alternative problem specification is to retain the ideal gas law density option, but use a reference density. This changes the source term calculation to:

S = ( - ref)g*Vol

If the reference density is evaluated at ambient temperature and pressure, then the zero pressure boundary condition is correct for this problem as well because there will be no source term for cells (or boundaries) at the reference conditions. Figure 4 shows the results for these conditions. Again, there is a downward velocity field with a small velocity magnitude. For this problem, the reference density was set equal to 1.21037999941. Using only six digits of precision gave velocity magnitudes of the order of 0.01m/s.

48

Flow Module

Figure 4. Solution using reference density

Proper Specification of Boundary Pressures

The final option is to retain all settings as in our original problem definition, but correctly specify the boundary pressures. Initially, a pressure of zero at the upper boundary, a pressure of gh at the lower boundary and an exponentially varying pressure along the vertical boundary. The pressure along the vertical boundary is exponential since the density varies with pressure rather than being constant.

What relaxation settings should I use? What is the difference between an Interial and Linear relaxation factor?

Under relaxation is a constraint on the change of a dependent or auxiliary variable from one solution iteration to the next. It is required to maintain the stability of the coupled, non-linear system of equations. The relax tab in the solver control panel (see Figure 1) allows the user to set under-relaxation factors for each of the solved variables and the auxiliary variables.

49


Figure 1. Solver Control - Under Relaxation

The panel contains four columns: the first defines the variable, the second contains a slider bar which can be used to adjust the value, the third contains up/down buttons to adjust the order of magnitude of the value, and the fourth is a field for the under relaxation value itself.

We have different methods for applying under relaxation for the solved and auxiliary variables that are listed below:

Inertial Relaxation (I):

Inertial under relaxation (I) is applied to variables, which are directly solved for (dependent variables as determined by active modules) during the iterative procedure, for example, velocities, pressure correction, enthalpy, etc.

I usually varies from 0.0 to 2.0 with default value of 0.2.

Increasing the value of I adds constraint. It means increasing I increases stability.

Increasing the value of I slows convergence. It means an increase in I will take more time to get the same order of convergence.

Values of I greater than 1.5 are allowed but not recommended.

Linear Relaxation (L):

Linear under relaxation (L) is applied to all variables that are computed during the solution procedure. These variables are called auxiliary variables, which are computed from the solved (dependent) variables, for example, density, pressure, temperature etc.

L usually varies from 0.0 to 1.0 with default value of 1.0.

Decreasing the value of L adds constraint. It means decreasing L increases stability.

Decreasing the value of L slows convergence. It means a decrease in L will take more time to get same order of convergence.

It all can be summarized in Figure-2 and Figure-3 below.

50

Flow Module

Figure 2. Under Relaxation for Faster Convergence

Figure-3. Under Relaxation for More Stability

Note: Please note that relaxation values can help in getting faster convergence or it may help prevent divergence. For a given problem (identical BC/VC/IC), change in relaxation values may take more or less number of iterations to reach convergence. But as long as problem is fully converged, you will get the same result irrespective of relaxation values.

Tips on troubleshooting your problems:

The following tips are just guidelines that can help in getting a converged solution or faster convergence. The values on relaxation can be problem specific, so there are no hard and fast rules as to which value one should use.

1. Problem Diverges:

If you see that your problem is diverging, you can try the following:

Make sure that you have applied correct scaling and all input values (BC/VC) are correct or at least in reasonable range.

Check the residual and see what variable diverges or starts diverging first.

Decrease the linear under relaxation (from 1.0 to say 0.7) for that variable.

In order to make it more stable, you can also increase inertial under relaxation for the corresponding solved (dependent) variable.

For compressible flows, decreasing linear under relaxation for density helps.

51


For problem involving heat transfer, if you see enthalpy is diverging, decreasing the linear relaxation of temperature from default value of 1.0 to smaller value like 0.7 can help in getting converged solution.

For Fluid Structure Interaction problems, decreasing the linear relaxation on pressure helps to moderate the pressure fluctuations seen by the stress solver, reducing the displacement fluctuations and aiding in convergence.

If it is a Fluid Structure Interaction problem and you encounter negative volumes, first try to decrease the linear relaxation for pressure to a value of 0.3 or 0.2. If the problems still exists, then you can try to decrease the linear relaxation for Grid Deformation anywhere from 0.5 to 0.1. This basically restricts the grid deformation in the solid volumes to 50% (if a value of 0.5 is used) of the actual value due to sustained pressures every time you solve for stress during the time step. Upon convergence, you still get the correct grid deformation.

For complex physics, when small changes in relaxation do not work, change the inertial relaxation values to 0.5. Also, reduce the linear factors to 0.3 and rerun. If this does not work, change the inertial factors to 0.9 and the linear ones to 0.1. These factors can be changed up to 1.5 for inertial and 0.01 for linear. Anything higher may result in a solution that has been frozen to the initial field.

Another item that may help is a change to the AMG solver for pressure correction or enthalpy. If convergence problems still persist, look at the residual information especially noting the location of the maximum residual. Next examine the grid closely at this spot in CFD-VIEW and look for skewness. Sometimes problem areas can be isolated by plotting the results every few iterations. The problem area is generally the location where the flow field first becomes unstable.

2. Slower Convergence:

If you see that convergence is very slow, you can try following:

Check the residuals and see what variable has slow convergence (might also remain flat)

Decrease the inertial under relaxation (from 0.2 to say 0.02) for that variable.

For conjugate heat transfer problems, decreasing the inertial relaxation of enthalpy from default value of 0.05 to smaller number like 1E-05 can help in faster convergence.

When solving for the electric module, decreasing the inertial relaxation of electric potential from 0.0001 to smaller number like 1E-07 can help in faster convergence.

How is the stream function calculated?

The stream function technique is useful for solving two dimensional flow problems. As an example, take two dimensional, incompressible flow in the x-y plane. For this situation, the stream function can be derived as follows:

This equation can be satisfied by introducing a stream function (x,y)such that:

therefore

52

Flow Module

Integrating this equation will yield the stream function. Lines of constant are streamlines of the flow, where d = 0.

In the case of steady simulation for a closed system which has no inlet/outlet, the pressure/density does not follow the surrounding wall temperature change, and the mass in the system is not conserved. How is this dealt with?

Basically, we make a correction to pressure to ensure continuity, not the density. For this case, there is no correction to pressure (grad(rho*V) = 0).

Details for a closed system in steady state with isothermal walls.

For a closed system in steady state with isothermal walls, if the temperature is doubled then one would expect the pressure to double. However, the density is reduced by a factor of two, which represents a mass loss. This is due to the fact that grad(rho*V) = 0, and thus the pressure does not change.

Flow Module Examples

The following tutorials use the Flow Module exclusively:

Laminar Flow Past a Backward Facing Step

Supersonic Flow over a Bump or Ramp

The following tutorials use the Flow Module in conjunction with one or more other modules:

Turbulent Flow Past a Backward Facing Step

Natural Convection between Concentric Thick-walled Cylinders

Oil Flow through a Compliant Orifice

Turbulent Mixing of Propane and Air (with and without reactions)

Transonic Flow Over NACA 0012 Airfoil

Multi-step Reaction in a Gas Turbine Combustor

Surface Reaction in a 2-D Reactor

Generic Semiconductor Reactor

Flow Module References

1. Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena. 2nd ed. New York: John Wiley & Sons, Inc, 2002. pp. 23-27, 84, 240-243, 848, 866.

2. Fung, Y.C., Biomechanics: mechanical properties of living tissues. 2nd ed., Springer, 1993.

3. P.D. Ballyk, D.A. Steinman and C.R. Ethier. Simulation of Non–Newtonian Blood Flow in an End–to–Side Anastomosis, Biorheology, Vol. 31: pp. 565–586, 1994.

4. Walburn, F.J. and D.J. Schneck, A Constitutive Equation for Whole Human Blood Biorheology, Vol. 13, pp. 201-210, 1976.

5. Giersiepen M., Wurzinger, L.J., Opitz, R., and Reul, H., “Estimation of Shear Stress-Related Blood Damage in Heart Valve Prostheses - in vitro Comparison of 24 Aortic Valves.” The International Journal of Artificial Organs 13.5(1990): 300-306.

53


54

6. Veijola, T., Kuisma, H., and Lahdenpera, J., “Equivalent Circuit Model of the Squeeze Gas Film in a Silicon Accelerometer.” Sensors and Actuators A 48(1995): 239-248.

7. W. Ostwald, Kolloid-Zeitschrift, 26, 99,-117 (1925); A. de Waele, Oil Color Chem. Assoc. J., 6, 33-38 (1923).

8. P. J. Carreau, Ph.D thesis, University of Wisconsin, Madison (1968).

Heat Transfer Module

Heat Transfer Module Introduction to the Heat Module

The Heat Transfer Module performs heat transfer analysis and is an integral part of the CFD-ACE-SOLVER. Use the Heat Transfer Module for all situations where heat transfer processes may have a significant impact on the final solution. Activating the Heat Transfer Module implies the solution of the total enthalpy form of the energy equation.

Many types of heat transfer analysis can be performed with the Heat Transfer Module, from basic conduction/convection to complex radiation modeling (with the use of the companion Radiation Module discussed in Radiation Module). Heat transfer analysis can be performed in stand-alone mode (pure heat transfer analysis) or coupled with other modules (such as the Flow, Mixing, Stress Modules, etc.) for a multi-physics simulation. The Heat Transfer Module includes:

Applications

Features

Theory

Limitations

Implementation

Examples

References

Heat Transfer Module Applications

CFD-ACE+ can simulate many types of heat transfer problems. The simplest are pure heat conduction problems (i.e., heat conduction through solids). More advanced applications will add the simulation of flow or mixing phenomena, and the most advanced will add higher physical models such as radiation and finite element stress solution. The Heat Transfer Module solves for the energy in the system and can be used to produce the temperature field and energy transfer characteristics of the model.

Thermal Field Calculations

The Heat Transfer Module is often used to determine the thermal field within a given geometry. Use CFD-ACE+ to predict the temperature field for comfort analysis or to determine if the materials can survive the temperature environment.

Heat Transfer Calculations

The Heat Transfer Module solves the energy (total enthalpy) equation, and can be used to determine the heat transfer characteristics of the system. CFD-ACE+ can determine the heat transfer rate through any boundary (internal or external) of the model. Heat transfer rate calculations help determine heating or cooling requirements.

Pure Conduction Problems The simplest heat transfer analysis problems are pure conduction problems. In these cases there is no fluid flow and all of the volume conditions are solids. CFD-ACE+ can handle these problems with ease and can simulate cases with multiple solids with different properties.

55


Conjugate Heat Transfer Problems

In many engineering problems, the flow domain consists of internal solids such as baffles, tubes, fins, or vanes. Thermal energy transport can occur across solid-fluid interfaces. In such cases, the fluxes on the solid and the fluid sides must match at the interface. This is the correct conservative way to solve the energy equation, and is called Conjugate Heat Transfer (CHT) analysis. Examples include cooling jacket flows, heat exchanger analysis, and ice-melting (defrosting) on windshields.

Natural Convection Problems

The Heat Transfer Module (in conjunction with the Flow Module) can be used to solve natural convection problems. These flows are seen in many applications, such as free convective cooling problems, and cooling towers.

Multi-Physics Applications

The Heat Transfer Module can be used with (and is required by) many of the other modules in CFD-ACE+ to perform multi-physics analyses. Some of the more commonly added modules are given in the list below. Examples of these types of applications are described in each module’s section.

Flow (with or without Turbulence)

Radiation

Mixing (with or without gas-phase and surface reactions)

Spray (with or without evaporation)

Stress

Plasma

Electrophysics

Biochemistry

Heat Transfer Module Features and Limitations of the Heat Module

Features

The Heat Transfer Module has the following built in features:

The ability to model ice melting problems

The ability to model moving (translating or rotating) solids (without grid motion)

The ability to model solidification problems

A special boundary condition to simulate a heat source adjacent to a wall

Ice Melting

The Ice Melting feature simulates the heat transfer requirements for the phase change of a material from a solid state to a liquid state. The solver, however, will not allow the material to flow after it has melted. This feature has been used extensively to simulate the transient defrosting process of automobile windshields.

Solidification

The Solidification feature simulates the heat transfer in phase change during the solidification process. Coupled with the flow module, this feature also allows you to simulate the mush flow in

56


the mushy zone. Two options are provided to describe the solidification process: isothermal and mushy.

Moving Solids

The Moving Solids feature simulates the heat transfer convection in a rotating or translating solid without the need for implicit grid motion. A volume condition can be selected to be moving so that the heat flow due to the motion of the solid can be captured. This feature has been used, for instance, to simulate the heating of translating parts in an oven, and the cooling of automotive disk brake rotors.

Wall Heat Sources

The Wall Heat Source feature is an additional heat source/sink that can be applied to the cells adjacent to wall boundary conditions. This allows the wall to be held to a fixed temperature, or heat flux, for instance while the adjacent cells are supplied with heat by other means (such as a thin (sub grid scale) strip heater or laser power deposition).

The wall, apart from being held at fixed temperature or heat flux, can also be held as adiabatic, external heat (convect), external heat (radiate), or external heat (both). Therefore, all the options under the Heat boundary conditions are applicable whenever you turn on the Wall Heat Source. See Boundary Conditions-Walls for details.

Limitations

The following limitations apply when using the Heat Transfer module:

When performing Ice Melting simulations, the solid known as ice is allowed to melt but is not allowed to flow after it has been melted.

Thin wall and parallel are not supported.

Total heat source feature cannot be used in parallel runs, use volumetric heat sources instead.

Arbitrary Interface on a conjugate wall is not recommended.

Heat Transfer Module Theory

The Theory section describes the mathematical equations used by the Heat Transfer Module. See Numerical Methods for details on the methods used to solve these equations.

Heat transfer processes are computed by solving the equation for the conservation of energy. This equation can take several forms and CFD-ACE+ numerically solves the energy equation in the form known as the total enthalpy equation. This form is fully conservative and is given in equation 2-1.[1]

(2-1)

where:

57


h o = the total enthalpy and is defined as:

(2-2)

where:

i

= The internal energy and is a function of the state variables and T

keff =

The effective thermal conductivity of the material. In laminar flow, this will be the thermal conductivity of the fluid, k. In turbulent flows:

where t is the turbulent Prandtl number.

p = The static pressure

ij = The viscous stress tensor which is described in detail in the Flow Module-Theory section

Sh contains terms for additional sources due to reactions, radiation, spray, body forces, etc. The actual source term for each of these features is described in the section that explains that feature.

Heat Module Model Setup

Heat Transfer Module Implementation and Grid Generation

The Heat Transfer section describes how to set up a model for simulation using the Heat Transfer Module of the CFD-ACE-Solver. The Heat Transfer Implementation section includes:

Model Setup and Solution - Describes the Heat Module related inputs to the CFD-ACE-Solver


You can apply the Heat Transfer Module to any geometric system (3D, 2D planar, or 2D axisymmetric). All grid cell types are supported (quad, tri, hex, tet, prism, poly).

The general grid generation concerns apply, for example, ensure that the grid density is sufficient to resolve thermal gradients, minimize skewness in the grid system, and locate computational boundaries in areas where boundary values are well known.

If you have regions in the geometry that have heat sources applied, or are moving solids or ice melting regions, then the regions where they exist must be separate volume conditions. Using CFD-GEOM terminology, in a structured grid, these will be separate structured blocks in 3D or faces in 2D. In an unstructured grid, the regions must be defined as separate unstructured domains in 3D or loops in 2D. This will help assign these regions as heat sources, ice melting, or moving solids.

Heat Transfer Module Problem Type

58



Select Heat Transfer to activate the Heat Transfer Module. The Heat Transfer Module is required for many simulations and can work in conjunction with most of the other Modules in CFD-ACE+. The only exceptions are the Cavitation and Free Surface Modules which must be run as isothermal problems.

Heat Transfer Module Model Options-Shared Tab

There are no settings under the Shared tab that directly affect the Heat Transfer Module. However, if free convection (buoyancy) flows are to be simulated then activation of the gravity source term under the Shared tab will be necessary. See Control Panel-Model Options for details about this option.

If you want to run a simulation using Chimera, simply activate the Chimera option then make the appropriate Chimera settings in the VC and BC tabs. For more information on the Chimera Grid option, please check the Chimera Grid Methodology chapter.

Heat Transfer Module Model Options-Heat Tab

The model options for the Heat Transfer Module are located under the Heat tab.

Model Options Panel in Heat Transfer Settings Mode

Ice Melting

Select Ice Melting to activate the ice-melting module. It is designed to compute the defrosting process for ice-build up on automobile windshields. The ice melting properties need to be provided as volume conditions and are described in Volume Conditions-Ice Melting Properties.

Solidification

Select Solidification to activate the solidification module that is designed to compute the solidification process and the phase change process. The solidification properties need to be provided as volume conditions and are described in Volume Conditions-Solidification Properties.

Moving Solid

59


Select Moving Solid to compute the convective terms in solids in the energy equation. Characteristics about the moving solids need to be provided as volume conditions and are described in Volume Conditions-Moving Solid Properties).

Heat Module Volume Conditions

Heat Transfer Module Volume Conditions

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. See Control Panel-Volume Conditions for details. Before any property values can be assigned, a volume condition entity must be made active by picking a valid entity from either the Viewer Window or the VC Explorer.

General heat sources can be specified by changing the volume condition setting mode to Heat. For additional information, see Numerical Methods-Discretization-Direct Specification of Source Terms, User Subroutines-User Defined Source Terms, and Solidification-Definition of Source Terms. With the volume condition setting mode set to Properties, select any volume conditions and ensure that the volume condition type is set to either Fluid or Solid.

There are two volume condition properties required by the Heat Transfer Module; specific heat and thermal conductivity.

If the Ice Melting feature has been activated, then inputs for the properties of the ice are required for solid volume conditions.

The methods used to evaluate the specific heat and conductivity properties and the required inputs are given in the table, Specific Heat Evaluation Methods and Required Inputs, and the following table, Conductivity Evaluation Methods and Required Inputs.

Specific Heat: Thermal Conductivity

Constant Constant User Subroutine (UCOND) Polynomial in T Prandtl Number Piecewise Linear in T Polynomial in T Mix JANNAF Method Piecewise Linear in T Mix Polynomial in T Mix Kinetic Theory Mix Piecewise Linear in T Mix Piecewise Linear in T User Subroutine (UCPH_FROM_T)

Mix Polynomial in T

Specific Heat

Constant

The constant options allows for the specification of the specific heat. This option is appropriate when the specific heat of the material does not depend on any other quantity, such as temperature.

Required Module (s): Heat Required Input (s): Specific Heat in J/kg-K

Polynomial in T

This option will calculate the specific heat as a function of temperature using a polynomial of the form:

60


Required Module (s): Heat Required Input (s): Polynomial Coefficients


The temperature and the corresponding specific heat at that temperature must be input, which the CFD-ACE-SOLVER will take and use to interpolate between values to set the specific heat. The interpolation is done as follows:

where n is the index for the table of inputs and runs from 1 to number of data pairs.

Required Module (s): Heat Required Input (s): Data pairs of Temperature and Specific Heat

Mix JANNAF Method

The mix JANNAF method is curve fits for calculating the specific heat and enthalpy of the following form

The coefficients are obtained from curve fits of experimental data.

Required Module (s): Heat, Chemistry Required Input (s): JANNAF coefficients, Lower temperature limit, Break point temperature, and Upper temperature limit

Mix Polynomial in T

The specific heat of the mixture is evaluated as

where

Required Module (s): Heat, Chemistry Required Input (s): Polynomial coefficients


61


The temperature and the corresponding specific heat at that temperature must be input for each species, which the CFD-ACE-SOLVER will take and use to interpolate between values to set the specific heat. The interpolation is done as follows:

where n is the index for the table of inputs and runs from 1 to number of data pairs. The specific heat for each species is calculated and the mixture specific heat is then evaluated as:

Required Module (s): Heat, Chemistry Required Input (s): Data pairs of Temperature and Specific Heat

User Subroutine (UCPH_FROM_T)

This option is available for implementing a user defined evaluation for specific heat if the option is not available through CFD-ACE-GUI. The user subroutine required for setting the specific heat is UCPH_FROM_T. For more information on user defined volume condition (property) routines, please refer to the volume condition routine section of the User Subroutines chapter.

Thermal Conductivity

Constant

The constant options allows for the specification of the thermal conductivity. This option is appropriate when the thermal conductivity of the material does not depend on any other quantity, such as temperature.

Required Module (s): Heat Required Input (s): Thermal Conductivity in W/m-K

Prandtl Number

This option allows for the specification of the Prandtl number, which CFD-ACE-SOLVER will then use to calculate the thermal conductivity. The thermal conductivity is then calculated as:

Required Module (s): Heat Required Input (s): Prandtl number

Polynomial in T

This option will calculate the thermal conductivity as a function of temperature using a polynomial of the form:

Required Module (s): Heat

62


Required Input (s): Polynomial Coefficients


The temperature and the corresponding thermal conductivity at that temperature must be input, which the CFD-ACE-SOLVER will take and use to interpolate between values to set the thermal conductivity. The interpolation is done as follows:

where n is the index for the table of inputs and runs from 1 to number of data pairs.

Required Module (s): Heat Required Input (s): Data pairs of Temperature and Thermal Conductivity

Mix Kinetic Theory[1]

The Mix Kinetic Theory option will use the kinetic theory of gases to calculate the thermal conductivity of the gas or mixture of gases. For a pure monatomic gas, the thermal conductivity is defined as

where

ki = thermal conductivity of species i

MWi = molecular weight of species i

T = temperature in Kelvin

i = characteristic diameter of the molecule in Angstroms

k = collision integral

The collision integral, k, is given by

where T* is the dimensionless temperature and is given by

where is the characteristic energy, is Boltzmann's constant, and T is the temperature. To calculate the mixture thermal conductivity using kinetic theory, the following equation is used:

where:

63


xi ,xj = mass fraction of species i and species j

i = viscosity of species i

= dimensionless quantity

and i,j is given by:

Required Module (s): Heat, Chemistry Required Input (s): Molecular Weight of each species, Characteristic Energy, and Collision Diameter. These quantities must be input in the Database Manager for each species.


The temperature and the corresponding thermal conductivity at that temperature must be input for each species, which the CFD-ACE-SOLVER will take and use to interpolate between values to set the thermal conductivity. The interpolation is done as follows:

where n is the index for the table of inputs and runs from 1 to number of data pairs. The specific heat for each species is calculated and the mixture specific heat is then evaluated as:

Required Module (s): Heat, Chemistry Required Input (s): Data pairs of Temperature and Specific Heat

Mix Polynomial in T

The thermal conductivity of the mixture is evaluated as

where

Required Module (s): Heat, Chemistry Required Input (s): Polynomial coefficients

64


User Subroutine (UCOND)

This option is available for implementing a user defined evaluation for thermal conductivity if the option is not available through CFD-ACE-GUI. The user subroutine required for setting the specific heat is UCOND. For more information on user defined volume condition (property) routines, please refer to the volume condition routine section of the User Subroutines chapter.

Heat Transfer Module Volume Conditions-Ice Melting Properties

If you have activated the Ice Melting feature, (see Heat Tab-Ice Melting for information on how to activate this feature), you will be able to set phase change properties for any solid type of volume condition. The default method is No Melting which means that the selected volume condition will not undergo a phase change calculation. If you change the evaluation method to Constant, you will be required to input the latent heat of fusion, melting temperature, and initial temperature. The solver will account for the energy required for the phase change process.

See Also

Volume Conditions-Solidification Properties

Volume Conditions-Moving Solid Properties

Heat Transfer Module Volume Conditions-Solidification Properties

If you have activated the Solidification feature (see Heat Tab-Solidification for information on how to activate this feature), you will be able to set phase change properties for any fluid type of volume condition. The default setting is No Solidification which means that the selected volume condition will not undergo a phase change calculation. If the Isothermal option is selected, you will be required to input the latent heat and solidification temperature.

If you choose the Mushy option, you must enter latent heat, melting temperature (TLow) and solidification temperature (THigh). The solver will account for the energy required for the phase change process. See Solidification Module-Solidification Process for details on the phase change process.

See Also

Volume Conditions-Ice Melting Properties

Volume Conditions-Moving Solid Properties

Heat Transfer Module Volume Conditions-Moving Solid Properties

If you have activated the Moving Solid feature, (see Heat Tab-Moving Solid for information on how to activate this feature) you can set the moving solid parameters for each solid type of volume condition.

To set moving solid parameters:

1. Set the volume condition setting mode to Heat.

2. Pick the volume conditions in the model that are of solid type.

3. Activate Moving Solid for the selected volume conditions.

4. Select a solid motion evaluation method (translation or rotation).

5. Specify the velocity (for translating solids) or rotation vector and center of rotation (for rotating solids).

See Also

65


Volume Conditions-Solidification Properties

Volume Conditions-Ice Melting Properties

Boundary Conditions

Heat Transfer Module Boundary Conditions-Introduction


The Heat Transfer Module is fully supported by the Cyclic, Thin Wall, and Arbitrary Interface boundary conditions. (See Cyclic Boundary Conditions, Thin-Wall Boundary Conditions, or Arbitrary Interface Boundary Conditions for details).

The boundary conditions for the Heat Transfer Module are located under the Heat tab and can be reached when the boundary condition setting mode is set to General. Each boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). This section describes the implementation of each type with respect to the Flow Module. The Boundary Conditions section includes:

Inlets/Outlets

Walls/Rotating Walls

Symmetry

Interfaces

Thin Walls

Cyclic

Heat Transfer Module Boundary Conditions-Inlets/Outlets

Inlets

There are no Heat Transfer Module related settings available for inlet boundary conditions, temperature at the inlet is specified under the Flow tab. See Inlets in Flow Module for more information.

Outlets

There are no Heat Transfer Module related settings available for outlet boundary conditions, temperature at the outlet is specified under the Flow tab. See Outlets in the Flow Module for more information. The specified outlet temperature will only be used in the case where there is inflow through the outlet boundary.

Heat Transfer Module Boundary Conditions-Walls/Rotating Walls

There are two types of wall boundary conditions available for the Heat Transfer Module:

The boundary condition itself (i.e., the computational boundary)

The ability to add a heat source to the cells adjacent to the wall boundary condition

For the wall boundary condition, the Heat Transfer Module needs to know how to set the heat flux for each cell face on the boundary condition patch. There are various ways to specify the

66


information and the following six methods (known as Heat Subtypes) are available when you click the Heat tab, and select one of the following from the Heat Subtype pull-down menu.

Adiabatic Option

The wall Adiabatic subtype sets the heat flux to zero. The wall temperature is allowed to float and will be calculated by the solver.

Isothermal Option

The wall Isothermal subtype enables you to set the wall temperature (Tw) to a specified value. The heat flux, qw, needed to maintain that value will be calculated by the solver as:

(2-3)

where:

k = fluid or solid conductivity

Tc = cell center temperature

dx = distance from the wall to the cell center

Heat Flux Option

The wall Heat Flux subtype enables you to fix the wall heat flux to a specified value. The wall temperature is allowed to float and will be calculated by CFD-ACE-SOLVER. When you select the Heat Flux subtype, the following panel appears and prompts you to select additional features:

Constant: a constant heat flux (W/m2) can be specified at this boundary

Profile X: you can input heat flux (W/m2) as a profile of X (m)

67


Profile Y: you can input heat flux (W/m2) as a profile of Y (m)

Profile Z: you can input heat flux (W/m2) as a profile of Z (m)

Profile 2D: you can input heat flux (W/m2) as a profile in a 2D plane

Profile in time: you can input heat flux (W/m2) as a profile in time (s)

Profile from file: you can input heat flux (W/m2) from an outside file (file name is required and the default path is current directory). The format of profile BC file can be found in Appendix A CFD-ACE+ Files.

Parametric: you can input heat flux (W/m2) as function of X, Y, Z, and T(time, s).

User Sub(ubound): heat flux (W/m2) will be specified by user subroutine. See User Subroutines (UBOUND).

External Heat Transfer (by convection) Option

The wall External Heat Transfer (Convect) subtype simulates heat transfer to/from the external environment (i.e., the area outside of the computational grid system) by convection. This subtype fixes neither the wall temperature or heat flux. Instead, the heat transfer at the wall is calculated as:

(2-4)

where:

hc = external heat transfer coefficient

Text = external temperature

The wall temperature (Tw) is determined by balancing the external and internal heat flux and solving for the wall temperature.

External Heat Transfer (by radiation) Option

The wall External Heat Transfer (Radiate) subtype simulates heat transfer to/from the external environment (i.e., the area outside of the computational grid system) by radiation. This subtype fixes neither the wall temperature or heat flux. Instead, the heat transfer at the wall is calculated as:

(2-5)

where:

= Stefan-Boltzmann constant (5.6696E-8 W/m2-K4)

e = external emissivity coefficient

= temperature of the radiation source or sink


68


External Heat Transfer (by convection and radiation) Option

The wall External Heat Transfer (Both) subtype combines the convection and radiation subtypes so that the heat transfer at the wall is calculated as:

(2-6)


Solid Cell at Wall

The Solide Cell at Wall option provides a simple treatment on the heat transfer in the external solid wall (no mesh is required in the wall). In the figures below, ic1 and ic2 are two cells neighboring an external wall boundary and solid cells are added in Figure 2. The key idea is described as:

The heat transfer between the wall and fluid cell:

(2-7)

Tw = wall temperature Tc = temperature at the cell center hf = fluid-side local heat transfer coefficient

Heat transfer from the solid cell to the fluid cell (no heat transfer from neighboring solid cells):

(2-8)

Ts = temperature of the solid cell ks = thermal conductivity of solid dn = thickness of the solid border

Wall Boundary - No Solid Cell Wall Boundary - With Solid Cell

For fixed flux and adiabatic wall boundary conditions:

69


(2-9)

With isothermal wall, Ts is fixed, then:

(2-9A)

High thermal conductivity in the solid will promote the heat conduction between the local boundary cell and its neighboring cells. The promotion effect can be included in the calculation of face conductivity.

Without solid cell, the face conductivity (between ic1 and ic2):

(2-10)

with solid cell:

(2-11)

As = face area between two neighboring solid cells Af = face area between neighboring fluid cells k = conductivity F = weight function

We recommend that this model be applied on smooth surfaces.

Wall Heat Source

When checked, this option enables you to have a heat source imposed on the cells adjacent to any wall boundary condition. Wall sources specify additional sources of heat in cells adjacent to wall boundary conditions while still maintaining the wall boundary condition as specified above (e.g., isothermal, adiabatic, etc.). Upon activating the Wall Heat Source option, you will be prompted to enter the per unit area heat source (W/m2) to be applied to all of the cells adjacent to the active wall boundary condition. This effectively adds a volumetric heat source to those cells. The total amount of heat added to each cell (W) will be the per unit area heat source value specified (W/m2) multiplied by the area of the cell’s boundary face (m2). Wall heat sources are not boundary conditions. They are additional conditions that are imposed at the wall and their value must be known prior to the calculation.

Heat Transfer Module Boundary Conditions-Symmetry

The symmetry boundary condition is a zero-gradient condition. Heat Transfer is not allowed to cross the symmetry boundary condition so it effectively behaves as an adiabatic wall. No values need to be specified for symmetry boundary conditions.

70


Heat Transfer Module Boundary Conditions-Interfaces

The interface boundary condition is used to allow two computational regions to communicate information. If the interface boundary condition is used to separate two solid regions, or to separate a solid and fluid region then a Wall Heat Source may be added. Interfaces that exist between two fluid regions cannot be used for Wall Source specification.

Interface boundary conditions can be converted to Thin Walls (see Thin-Wall Boundary Conditions). See Arbitrary Interface Boundary Conditions for information on other ways for computational domains to communicate.

Heat Transfer Module Boundary Conditions-Thin Walls

The Thin Wall boundary condition is fully supported by the Heat Transfer Module. You may optionally choose to activate the Thermal Gap Model feature which reduces the heat transfer across the thin wall. (See Thin-Wall Boundary Conditions for instructions on how to setup a Thin Wall boundary condition.

There are two Heat Transfer Module related settings available for a thin wall boundary condition; thickness, and conductivity. Both of these settings can be found under the Heat Transfer (Heat) tab when the thin wall has been selected.

The thickness and conductivity settings enable you to impose a heat transfer resistance which causes a temperature jump to be calculated across the thin wall. See Thin-Wall Boundary Conditions for details on how to set these values.

You may choose to activate the Wall Source feature which adds a heat source to the cells adjacent to each side of the thin wall boundary condition. The wall source feature is described in detail in Walls and its application to thin walls is described in Thin-Wall Boundary Conditions.

Heat Transfer Module Boundary Conditions-Cyclic

The Cyclic boundary condition is fully supported by the Heat Transfer Module. See Cyclic Boundary Conditions for instructions on how to setup a Cyclic boundary condition. There are no Heat Transfer Module related settings for the Cyclic boundary condition.

Heat Transfer Module Initial Conditions

Click the Initial Conditions [IC] tab to see the Initial Condition Panel. See Control Panel-Initial Conditions for details.

The Initial Conditions can either be specified as constant values or read from a previously run solution file. If constant values are specified then you must provide initial values for the Heat Transfer Module. The only value that needs to be specified is temperature (T).

Although the Initial Condition values do not affect the final solution, you should specify reasonable values so that the solution does not have convergence problems at start-up.

Heat Transfer Module Solver Control Settings

Spatial Differencing Scheme

Under the Spatial Differencing tab you can select the differencing method to be used for the convective terms in the equations. Activating the Heat Transfer Module enables you to set parameters for enthalpy calculations. The default method is first order Upwind. See Control

71


Panel-Spatial Differencing Scheme for details on the different differencing schemes. See Numerical Methods for numerical details of the differencing schemes.

Solver Selection

Under the Solvers tab, select the linear equation solver to be used for each set of equations. Activating the Heat Transfer Module enables you to set parameters for enthalpy calculations. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 50 sweeps. The default convergence criteria is 0.0001. See Control Panel-Solver Controls for more information on the different linear equation solvers available. Also see Linear Equation Solvers for numerical details of the linear equation solvers.

Relaxation Parameters

Under the Relaxation tab, select the amount of under relaxation to be applied for each of the dependent (solved) and auxiliary variables used for the energy equation. Activating the Heat Transfer Module enables you to set parameters for the dependent variable enthalpy, as well as the auxiliary variable, temperature. See Control Panel-Under Relaxation Parameters for details on setting the under relaxation values. See Numerical Methods-Under Relaxation for numerical details of how under-relaxation is applied.

The enthalpy equation uses an inertial under relaxation scheme and the default value is 0.2. Increasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.

The calculations for temperature use a linear under relaxation scheme and the default values are 1.0. Decreasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.

The default values for all of the under relaxation settings will often be sufficient. In some cases, these settings will have to be changed, usually by increasing the amount of under-relaxation that is applied. If the heat transfer problem is fairly simple, then the inertial factor for enthalpy can often be reduced to allow faster convergence. There are no general rules for these settings and only past experience can be a guide.

Variable Limits

The Limits Tab enables you to set minimum and maximum variable values. CFD-ACE+ will ensure that the value of the variable will always remain within these limits by clamping the value. Activating the Heat Transfer Module enables you to set limits for enthalpy and temperature variables. See Control Panel-Variable Limits for details on how limits are applied.

Advanced Settings

In CFD-ACE+, by default, inertial under-relaxation of dependent variables is used to constrain the change in the variable from one iteration to the next in order to prevent divergence of the solution procedure.

You can switch the default inertial relaxation method to the CFL based relaxation method by going to the Solver Control panel's Advanced tab and checking the appropriate check boxes for each module.


The relaxation factor defined in SC-->Relax is used as the CFL multiplier. A general rule would be the inverse value of usual inertial relaxation factor.

Effect of Value:

5 = Default Value


72



Viscous Dissipation - heating due to viscous work by the fluid, i.e. friction heating.

Heat Transfer Module Output Options

The desired output can be specified under the Out (Output) tab. There are two types of output available, Printed and Graphical. Printed output will be written to a text file, whether it be the modelname.out file or another file. Graphical output will be written and stored in the DTF file and available for further post-processing in CFD-VIEW.

Output

For steady state simulations, there are two options for when results will be written to the DTF file. The first is End of Simulation. This option will only write results to the DTF file once the maximum number of iterations has been reached or when the specified convergence criteria is achieved. The Specified Interval option allows for results to be written at certain intervals during the solution process. The file will be uniquely named in the following way: modelname_steady.000025.DTF. A unique file can be created or the results can be written to the same DTF file, modelname.DTF.

For transient simulations, the results will be written out at the specified time step frequency.

Printed Output

Under the Print tab, select the printed information to be output to the text based output file (modelname.out). Activating the Heat Transfer Module enables the output of a heat transfer summary in addition to the general printed output options. See Control Panel-Printed Output for details on printed output options including boundary condition integral output, diagnostics, and monitor point output.

Selecting the Heat Flux Summary option provides a Boundary-by-Boundary Heat Transfer Summary printout of the integrated heat transfer (W (3D), W/m (2D), or W/rad (axisymmetric)) through each of the thermal boundaries (walls, inlets, outlets, interfaces, etc.). The fields include:

Name - Lists the name of the boundary.

Key - The key number associated with each boundary.

Type - Lists the boundary type: walls, inlets, outlets, interfaces, etc.

COND. +CONV. - Includes any heat source/sink due to conduction and convection.

W_SRC. +CVD - W_SRC refers to any heat source/sink that is applied on the BC - Heat tab for a wall boundary condition. The source is applied to the cells directly adjacent to the wall. CVD refers to any source/sink of heat due to a surface reaction.

RAD to SYS* - Any heat source/sink due to radiation.

Sum - Summation of all the heat sources/sinks over all boundaries.

Total pressure work - This is the rate of work done on the fluid per unit volume by pressure forces

Total volume source - This reports the data pertaining to a volume heat source in the system, if present. Such a heat source can be introduced in the system during problem set up in CFD-ACE-GUI under VC VC Setting Mode:Heat.

Transient term - This term only appears for transient cases. The transient term is the amount of heat that is entered the system at the current time step, but has not yet left the system.

For radiation cases, a radiative heat summary is printed in the output file when the Heat Transfer summary is activated on the Printed Output tab. This summary reports the radiative heat into for

73


all boundaries in the model. If Monte Carlo radiation is used, then the patch type, patch temperature, and the number of rays absorbed is also reported. The format is:

Name - Lists the name of the boundary

Key - The key number associated with each boundary

Type - Lists the boundary type: walls, inlets, outlets, interfaces, etc.

EMIT - Amount of heat being emitted from the boundary

ABSORB - Amount of heat being absorbed by the boundary

RAD Net - Net radiative heat on the boundary (RAD Net = ABSORB -EMIT)

RAD to SYS* - Amount of heat contributed to the system

Graphical Output

Under the Graphics tab, you can select which variables to output to the graphics file (modelname.DTF). These variables will then be available for viewing and analyzing in CFD-VIEW. Activating the Heat Module enables output of the variables listed in the table:

Heat Transfer Module Graphical Output

Variable Units

Static Temperature K

Total Temperature K

Static Enthalpy J/kg

Specific Heat J/kg-K

Conductivity W/m-K

Wall Heat Flux W/m2

Heat Residual -

Heat Transfer Module Post Processing

CFD-VIEW can post-process solutions. When the Heat Transfer Module is invoked, the temperature field is usually of interest. A list of Heat Module post processing variables is shown in the table below. You can view the temperature field with surface contours and analyze it using point and line probes.


74



COND Conductivity W/m-K

CONDX X-direction Conductivity

W/m-K

CONDY Y-direction Conductivity

W/m-K

CONDZ Z-direction Conductivity

W/m-K

CP Specific Heat J/kg-K

H0 Total Enthalpy m2/s2

T Temperature K

T_TOT Total Temperature K

Wall_Heat_Cond_Flux Wall Heat Flux W/m2

Wall_Heat_Rad_Flux Wall Radiative flux W/m2

The heat transfer summary written to the output file (modelname.out) is often used to determine quantitative results and judge the convergence of the simulation. Due to the law of conservation of energy, the summation of all heat transfer into and out of the computational domain should be zero (unless heat sources or sinks are present). In the simulation a summation of exactly zero is almost impossible, but you should see a summation that is several orders of magnitude below the total heat transfer into the system.

Heat Module Frequently Asked Questions

What is the Viscous Dissipation option?

Due to shear stresses in a flowing fluid, one layer of a fluid “rubs” against an adjacent layer of fluid. This friction between adjacent layers of the fluid produces heat; that is, the mechanical energy of the fluid is degraded into thermal energy. The resulting volumetric heat source is called viscous dissipation.

In most flow problems viscous dissipation heating is not important. However, this heating can produce considerable temperature rises in systems with large viscosity and large velocity gradients. Examples of situations where viscous heating must be accounted for include: (i) flow of a lubricant between rapidly moving parts, (ii) flow of highly viscous fluids in high-speed viscometers, and (iii) flow of air in the boundary layer during rocket reentry problems.

A non-dimensional number called the Brinkman number is a measure of the importance of the viscous dissipation term. The Brinkman number, Br, is given by the ratio of the viscous heating to the conductive heating.

where

75


76

�= fluid velocity

�= fluid viscosity

k = fluid thermal conductivity T = fluid temperature T0 = reference temperature

Typically, the viscous dissipation must be taken into account if the calculated Brinkman number has a value greater than 0.1.

In the CFD-ACE-GUI, viscous dissipation is on by default under the SC/Adv/Heat Transfer section when the Heat module is activated. It can be deactivated if the Brinkman number shows that viscous heating is negligible for the model of interest.

Heat Transfer Module Examples

The following tutorials use the Heat Transfer Module exclusively:

Conduction between Concentric Thick-walled Cylinders

The following tutorials use the Heat Transfer Module in conjunction with one or more other modules:

Natural Convection between Concentric Thick-walled Cylinders


Oil Flow through a Compliant Orifice


Surface Reaction in a 2D Reactor

Generic Semiconductor Reactor

Heat Transfer Module References

Versteeg HK and Malasekera, W., "An Introduction to Computational Fluid Dynamics." John Wiley & Sons, Inc. New York, 1995. pp20.

Turbulence Module

Turbulence Module Introduction

The Turbulence Module enables you to simulate the effects of turbulence. Turbulence may have strong influence on momentum, heat, and mass transfer. For problems with high values of Reynolds numbers, you must activate the Turbulence Module and select an appropriate turbulence model to simulate the effect of turbulence on the mean flow.

There are two main methods for studying turbulent flows: Reynolds Averaged Navier-Stokes simulations (RANS) and Large Eddy Simulations (LES). CFD-ACE+ offers a wide choice of RANS and LES turbulence models in the CFD-ACE-SOLVER. In all these models, the effect of turbulence on transport is accounted for via turbulent or eddy viscosity. The Turbulence Module includes:

Turbulence-Applications

Turbulence-Features

Turbulence-Theory

Turbulence-Limitations

Turbulence-Implementation

Turbulence-Frequently Asked Questions

Turbulence-Examples

Turbulence-References

Turbulence Module Applications

Turbulent flow is encountered in a large number of practical applications in various industries, including, but not limited to, turbo machinery, aerodynamic engineering, automotive engineering, and civil engineering. Any moderate to high Reynolds number flow problem will involve turbulence.

Turbulence Module Features

The CFD-ACE-Solver has several built-in turbulence models available. These models are explained in detail in the Turbulence Theory section.

Standard k- Model

RNG k- Model

Kato-Lauder k- Model

Low Reynolds Number k- Model (Chien)

Two-Layer k- Model

k- Model

k- SST Model

77


Spalart-Allmaras Model

Large Eddy Simulations - SGS Models

o Smagorinsky Model

o Germano's Dynamic Subgrid Scale Model

o Menon's Localized Dynamic Subgrid Scale Model

Constant Turbulent Viscosity

User Defined Turbulent Viscosity

Turbulence in Porous Media

The Turbulence Module can add surface roughness effects to the turbulence model through the input of a roughness height. Details of this feature are in Turbulence-Boundary Condition.

Turbulence-Theory

Turbulence Module Theory-Introduction

For more than a century the preferred approach in the treatment of turbulent flows is to predict macroscopic statistics using the RANS formalism. Introduced by Reynolds in 1895, it involves a simple decomposition of the instantaneous fields in mean values and fluctuations via an averaging operation. The issue of turbulence modeling arises from the need to represent turbulent or Reynolds stresses, which are additional unknowns introduced by averaging the Navier-Stokes equations. A common approach adopted by CFD-ACE+ is the Eddy Viscosity approximation in which the Reynolds stress tensor is assumed to be proportional to the rate of mean strain, by analogy with the laminar stress-strain relationship. The proportionality parameter is called the turbulent or eddy viscosity, and is expressed phenomenologically or obtained from transport equations. Unlike its laminar counter-part, the turbulent viscosity is not a property of the fluid but rather a characteristic of the flow.

Within the framework of RANS modeling, various models differ in the way the turbulent viscosity is calculated. These models are typically categorized by the number of additional transport equations to be solved. Almost all the models in CFD-ACE+ involve solutions of two extra transport equations. One is for the turbulent kinetic energy, k, and the other is for the rate of dissipation, , or the specific rate of dissipation, .

Based upon the way the near-wall viscous sublayer is handled, these models are further classified into high-Reynolds-number and low-Reynolds-number models. Here the qualifier Reynolds-number refers to the local turbulent Reynolds number:

(3-1)

It will be shown that Ret is proportional to the ratio of the eddy viscosity to molecular viscosity, v. High Reynolds models are designed for regions where the eddy viscosity is much larger than the molecular viscosity and, therefore, cannot be extended into the near-wall sublayers where viscous effects are dominating. The standard wall-function model is used to bridge the gap between the high-Re-number regions and the walls or to connect conditions at some distance from the wall with those at the wall. Low-Reynolds-number models are designed to be used in the turbulent core regions and the near-wall viscous sublayers.

78

Turbulence Module

CFD-ACE+ contains several different turbulence models. You can choose any one of them to calculate the turbulent viscosity. Mathematical formulations of all models are described in the Theory section:

Theory-Reynolds Averaged Navier-Stokes Simulations

o Standard k-e Model

o RNG k-e Model

o Kato-Launder k-e Model

o Low Reynolds Number k-3 Model (Chien)

o Two-Layer k-e Model

o k-w Model

o k-w SST Model

o Spalart-Allmaras Model

Theory-Large Eddy Simulations

Theory-Constant Turbulent Viscosity

Theory-User Defined Turbulent Viscosity

Theory-Turbulence in Porous Media

Reynolds Averaged Navier-Stokes Simulations

Turbulence Module Theory-Standard k- Model

Several versions of the k- model are in use in the literature. They all involve solutions of transport equations for turbulent kinetic energy and its rate of dissipation. The one adopted in CFD-ACE+ is based on Launder and Spalding (1974). In the model, the turbulent viscosity is expressed as:

(3-2)

The transport equations for k and are,

(3-3)

(3-4)

with the production term P defined as:

(3-5)

79


The five constants used in this model are:

The standard k- model is a high Reynolds model and is not intended to be used in the near-wall regions where viscous effects dominate the effects of turbulence. Instead, wall functions are used in cells adjacent to walls.

Adjacent to a wall the non-dimensional wall parallel velocity is obtained from

(3-6)

(3-7)

where:

Here yv+ is the viscous sublayer thickness obtained from the intersection of equation 3-6 and

equation 3-7. The production and dissipation terms appearing in the turbulent kinetic energy transport equation are computed for near wall cells using (Ciofalo and Collins, 1989):

(3-8)

(3-9)

Similarly for heat transfer if we define a non-dimensional temperature,

80

Turbulence Module

(3-10)

then the profiles of temperature near a wall are expressed as (Ciofalo and Collins, 1989):

(3-11)

(3-12)

where P+ is a function of the laminar and turbulent Prandtl numbers ( and t) given by Launder and Spaulding (1974) as:

(3-13)

Here yT+ is the thermal sublayer thickness obtained from the intersection of equation 3-11 and

equation 3-12. Once T+ has been obtained, its value can be used to compute the wall heat flux if the wall temperature is known, or to compute the wall temperature if the wall heat flux is known.

Turbulence Module Theory-RNG k- Model

A variation of the k- model was developed by Yakhot and Orszag (Yakhot and Orszag, 1986) using a renormalization group (RNG) approach in which the smallest scales of motion are systematically removed. This model was subsequently modified by Yakhot et. al.

(1992). The model is formulated such that the equations for k and (equation 3-13 and equation 3-14) have the same form as the standard k- models. The model coefficients, however, take different values as:

The coefficient C1 becomes a function of , the ratio of time scales for turbulence and mean strain rate.

(3-14)

(3-15)

The constants in equation 3-14 have the values 0=4.38 and =0.015. The rate of mean-strain tensor, Sij, is defined as follows:

81


(3-16)

The RNG k- turbulence model is a high Reynolds number model, so the k and equations are not integrated to the wall. Wall functions, described in Standard k- Model, specify the values of k and at boundaries.

Turbulence Module Theory-Kato-Launder k- Model

Another extension to the standard k- model was given by Kato and Launder (1993) in their study of turbulent flow around bluff bodies. They found it necessary to modify the turbulence production term to reduce the excessive level of turbulence, given by the standard model, in regions of flow stagnation. To illustrate the modification, we first recast the standard production term as:

(3-17)

where Sij is the strain tensor as defined in equation 3-16 (see RNG k- Model).

In the Kato-Launder model the production term is modified as:

(3-18)

where ij is the vorticity tensor and is defined as:

(3-19)

The Kato-Launder k- turbulence model is a high Reynolds number model, so the k and equations are not integrated to the wall. Wall functions, as described in the Standard k- Model section, are used to specify the values of k and at boundaries.

Turbulence Module Theory-Low Reynolds Number k- Model (Chien)

High Reynolds number k- models require the use of wall functions. However, the commonly used wall functions may not be accurate in flows, which include phenomena such as large separation, suction, blowing, heat transfer, or relaminarization. This difficulty associated with wall functions can be circumvented by using low-Reynolds number k- models that permit the integration of momentum and k- equations all the way to the wall. Several versions of low-Reynolds number k- models have been proposed. The k- equations are modified to include the effect of molecular viscosity in the near wall regions. The general form of low-Reynolds number k- models is given by the following equations:

82

Turbulence Module

(3-20)

(3-21)

(3-22)

The low Reynolds model of Chien (1982) has been implemented in CFD-ACE+. The model parameters appearing in the preceding equations are:

(3-23)

Since the wall shear stress is computed from finite differences for this model, the first grid-point should be placed in the laminar sublayer (y+ ~ 1). Therefore, the Chien model requires the use of very fine grids near solid boundaries.

Turbulence Module Theory-Two-Layer k- Model

The difference between the high and low-Reynolds number models lies in the near-wall treatment. With wall-function approaches, the high-Reynolds number models are, computationally, more robust and cost-effective. However, such near-wall treatment only provides fair predictions of skin friction when the flow runs primarily parallel to the wall and when the adjacent-to-wall grid cell center lies above the viscous layer, say, y+ > 11.5. In the presence of complex geometry and flow conditions, wall-functions lose a considerable amount of accuracy. On the other hand, low-Reynolds number models may yield more accurate results but require extensive grid refinement near the wall and are thus more expensive to use.

As a comprise, the concept of two-layer modeling is introduced (Chen & Patel, 1988) in which the near-wall sublayer is divided into two layers. We use the standard k- model in the outer layer where turbulent effect dominates. In the inner layer where viscous effect prevails, we use a one-equation model where the -equation is replaced by an algebraic relation. With the two-layer model, turbulent viscosity is calculated as:

83


(3-24)

The damping function f is defined as:

(3-25)

and the length scales are defined as:

(3-26)

(3-27)

where the local turbulent Reynolds number is defined as:

The model constants are a = 50.5, b = 5.3, Cl = C-3/4 and the interface location is at f= 1, below

which the rate of dissipation is calculated as:

(3-28)

Turbulence Module Theory-k- Model

The k- turbulence model is a two-equation model that solves for the transport of , the specific dissipation rate of the turbulent kinetic energy, instead of . The k- model in CFD-ACE+ is based on Wilcox (1991). The eddy viscosity in this model is:

(3-29)

where:

(3-30)

The transport equations for k and are:

84

Turbulence Module

(3-31)

(3-32)

The model parameters in the above equations are all assigned constant values:

(3-33)

The boundary conditions for k and at wall boundaries are:

k = 0 at y = 0, (3-34)

(3-35)

where y1 is the normal distance from the cell center to the wall for the cell adjacent to the wall. The location of the cell center should be well within the laminar sublayer for best results (y+ ~ 1). This model, therefore, requires very fine grids near solid boundaries.

Turbulence Module Theory-k- SST Model

Menter transformed the standard k- model into the k- form, and developed a blending function

that is equal to one in the inner region and goes gradually towards zero near the edge of the boundary layer(Menter, 1994). In the inner region the original k- model is solved, and in the outer region a gradual switch to the standard k- model is performed. The idea behind the SST model is to introduce an upper limit for the principal turbulent shear-stress in the boundary layers in order to avoid excessive shear-stress levels typically predicted with Boussinesq eddy-viscosity models.

The SST model performs like k- model, but less sensitive to the free-stream values, .

85


where is the rotation and curvature sensitization function.

The model constant are evaluated from:

= 0.41, and can be calculated from:

the last term in equation is the cross-diffusion term, which makes the model insensitive to the

free-stream . The blending function, , is given by:

and

stands for the positive portion of the cross-diffusion

is obtained from the maximum value of cross-diffusion term multiplied by a factor of

.

86

Turbulence Module

is defined as

where = 0.31, is a scalar measure of the vorticity tensor which is defined by:

This is the SST limitation for . The purpose of function is to prevent the activation of the SST limitation in the free shear flows. It is given by

The function is designed to prevent the SST limitation from being activated in the roughness layer in rough-wall flows, and it is given by [Antti, 1997]

d is the distance to the nearest surface point.

Turbulence Module Theory-Spalart-Allmaras Model

The Spalart-Allmaras model is a one-equation model that solves a transport equation for the kinematic eddy viscosity (1992). This model has been specifically designed for aerospace applications. CFD-ACE+ uses a wall function approach and solves the following transport equation for the eddy viscosity:

(3-36)

where:

87


The transport equation is solved using the following model constants:

Turbulence Module Theory - V2F Model

In the model, the standard equations are employed:

(1)

(2)

with the wall condition:

(3)

The rate of turbulence kinetic energy production is defined as:

88

Turbulence Module

(4)

The model is supplemented with two additional equations, the first for a velocity scale :

(5)

where the factor is obtained from:

(6)

A pseudo-time derivative is used for solving the equation in order for the transport equations to have the same structure as the other model equations. The time scale and length scale are given by:

(7)

In the model which yields the wall conditions

(8)

The eddy viscosity is computed from:

(9)

The model constants appearing above are:

(10)

Turbulence Module Theory - Modified V2F Model

A modified version of the model was also applied to prediction of the flow over the ramp. The modified form is that in which a simple modification is used to enhance the numerical robustness of the overall approach. In codes with explicit and/or uncoupled advance (i.e. for which the model equations are advanced following the momentum equations) a numerical

instability can occur due to the term in the denominator of the model equation where . To alleviate this problem, a modification is made by setting and in this way it is

89


possible to apply the condition as the wall boundary condition. The other constants which change are and with the remaining constants having the same values

outlined before. For all the models the derivatives of , , , and with respect to the streamwise coordinate are set to zero at the outflow boundary. At each iteration the transport

equation for , , , and are solved using successive over-relaxation until the maximum residuals reduce to machine zero.

Large Eddy Simulations

Turbulence Module Theory-Large Eddy Simulations Introduction

Large Eddy Simulations (LES) are considered somewhere between the model-free Direct Numerical Simulations (DNS) and RANS with respect to both physical resolution and computational costs. LES partly inherits the robustness and universality of DNS, allowing accurate prediction of the coherent structures in turbulent flows. The cost for LES is lower than for DNS because the resolution requirements for LES are of the same order as those for RANS. In most turbulent flows of practical interest the motion on the order of the dissipation scale cannot be evaluated explicitly due to limitations on the available computational resources required to resolve the physics of the flow. To overcome this limitation, the governing equations have to be altered in such a way that the activity at the level of unresolved scales is mimicked by a proper model, and only the large-scale fluctuations are explicitly taken into account.

In LES, a smoothing (low pass) filter of constant kernel width achieves separation of scales, decomposing a given field into a resolved component and a residual component (also called sub-grid fluctuations).

Operationally, the filtering is described by the convolution:

(3-37)

where represents the filtered value of the field variable f, G denotes the filter, which is a symmetric function with compact support andf is the filter width (assumed constant in the standard LES formulation). In variable density flows, it is best to use Favré (density weighted) filtering. Applying the filtering operation, the Navier-Stokes equations for the evolution of the large-scale motions are obtained. The filtered equations contain unknown terms such as

(velocity-velocity correlation) arising from the filtering of nonlinear terms and are known as subgrid scale (SGS) stresses.

Turbulence Module Theory-Large Eddy Simulations-SGS Models Introduction

The most popular model for engineering applications is arguably the Smagorinsky model (1963), where the eddy viscosity is proportional to the square of the grid spacing and the local strain rate. The constant of the model follows from an isotropy-of-the-small-scales assumption. The standard Smagorinsky model gives interesting results in free-shear flows, but fails in the presence of the boundaries and is proverbial nowadays for its excessive dissipation. Attempts to determine the model constant in a flow dependent fashion, have produced several generations of the dynamic model since the paper of Germano, (1992). Using a double filtering technique, the constant arising in the Smagorinsky model is computed as a function of space and time.

Turbulence Module Theory-Large Eddy Simulations-SGS Models-Smagorinsky Model

90

Turbulence Module

Based on the original SGS model proposed by Smagorinsky (1963), the SGS eddy viscosity is computed based on the grid spacing and local strain rate:

(3-38)

where is the filter (grid) width. The constant (Cs = 0.05-0.2) of the model follows from the isotropy-of-the-small-scales assumption.

Turbulence Module Theory-Large Eddy Simulations-SGS Models-Germano's Dynamic Subgrid-Scale Model

The Smagorinsky model constant is dynamically determined local flow conditions (Lilly et al. 1992). The grid-filtered Navier-Stokes equations are filtered again using a test filter larger than the grid size and eddy viscosity is computed as:

(3-39)

where is the filter width and Lij and Mij are related to sub-grid and sub-test filter scale stresses (Galperin & Orzag 1993). In addition to the strain invariant, the dynamic model requires the computation of filtered velocities, Reynolds stresses, and strain components and consumes more computational resources.

Turbulence Module Theory-Large Eddy Simulations-SGS Models-Menon's Localized Dynamic Subgrid-Scale Model (LDKM)

The LDKM model uses scale-similarity and the subgrid-scale kinetic energy:

(3-40)

to model the unresolved scales. Using ksgs the SGS stress tensor is modelled as:

(3-41)

with the resolved-scale strain tensor defined as:

(3-42)

In the modelling of the SGS stresses, implicitly the eddy viscosity is parameterized as:

91


. (3-43)

The subgrid-scale kinetic energy is obtained by solving the transport equation:

.

(3-44)

which is closed by providing a model for the SGS dissipation rate term, sgs based on simple scaling arguments:

(3-45)

In these models, C and C are adjustable coefficients determined dynamically using the information from a resolved test-scale field. The test-scale field is constructed from the large

scale field by applying a test filter which is characterized by , the test filter width. In this project, with arbitrary grids, we are using a test filter consisting of a weighted average of the cells sharing a node with the current cell. This average is biased towards the current cell, with a weight equal to the number of vertices of the cell. The cells that share a face with a current cell have a weight of two.

The application of the test filter on any variable is denoted by the top hat. By definition, the Leonard stress tensor at test-scale level is:

. (3-46)

The Leonard stress tensor and the SGS tensor are known to have high degrees of correlation, which justifies the use of similarity in the derivation of the dynamic model coefficients. The resolved kinetic energy at the test filter level is defined from the trace of the Leonard stress tensor:

.

(3-47)

This test scale kinetic energy is dissipated at small scales by:

.

(3-48)

Based on a similarity assumption and using appropriately defined parameters, the Leonard stress tensor has a representation analogous to the SGS stress tensor:

92

Turbulence Module

.

(3-49)

The least square method is applied to obtain the model constant:

.

(3-50)

where:

.

(3-51)

Finally, a corresponding approach is used to determine the dissipation rate constant. By invoking similarity between the dissipation rates at the subgrid level and at the test scale level C is determined to be:

.

(3-52)

The coefficients of the LDKM model are Galilean invariable and realizable. This model is also quite simple and efficient, does not rely on ad hoc procedures, and is applicable to various flow fields without adjustment of the model.

Transition Models

Turbulence Theory Transition Models Introduction

This model is based on two transport equations: an intermittency equation used to trigger the transition process, and an equation for transition momentum thickness Reynolds number used as the local onset parameter. The strength of this model is to mimic the quality of correlations based model but put in the framework of local (single point) approach so that it is realizable within a modern, multi-domain CFD framework.

A disputable drawback of this model is the added computational requirements, since it solves two additional transport equations. The additional CPU requirement, however, can be reduced by solving only the intermittency equation using a user specified value of the transition onset Reynolds number ( is treated as a constant).

Turbulence Theory Transport Equation for Intermittency

The intermittency equation is formulated as follows:

(1)

93


The source terms are defined as

(2)

(3)

where is the strain rate magnitude. , which serves to trigger the intermittency production, is formulated as a function of the vorticity Reynolds number,

(4)

(5)

(6)

(7)

(8)

(9)

where the critical Reynolds number where the intermittency first starts to increase in the boundary

layer, in equation 5, is related to the transition momentum thickness Reynolds number according to

(10)

with function in such a way that occurs upstream of . The function in equation 2 is an empirical correlation that controls the length of the transition region.

The destruction terms are defined as

94

Turbulence Module

(11)

(12)

where is the vorticity magnitude. These terms ensure that the intermittency remains zero in the laminar boundary layer (it is one in the freestream) and also enables the model to predict re-laminarisation. The function is designed to disable the destruction terms outside of a laminar boundary layer or viscous sublayer, and is defined as

(13)

The model constants for the intermittency equation are

(14)

(15)

(16)

The boundary condition for at a wall is zero normal flux while at inlet is equal to 1. For accurate

transition the grid must satisfy . If is too large (5), the transition onset location moves

upstream with increasing . The recommended advection scheme is second order upwind or central scheme, and the wall normal grid expansion ratio is about 1.1. The correlation functions

and at present are proprietary to CFX. The following correlations have been proposed to replace the proprietary versions,

(17)

(18)

Turbulence Theory Transport Equation for Transition Momentum Thickness Reynolds Number

The The transport equation for the transition momentum thickness Reynolds number is defined as follows:

95


(1)

The source term is designed to force the scalar to match the local value of calculated from an empirical correlation outside the boundary layer

(2)

(3)

with being a time scale for dimensional reason. The blending function is used to turn off the source term in the boundary layer. It is thus equal to zero in the freestream and one in the boundary layer

(4)

with

(5)

The model constants for the onset Reynolds number equation are

(6)

The boundary condition for at a wall is zero flux, while at an inlet should be calculated from the empirical correlation based on the inlet turbulence intensity.

Turbulence Theory Correction for Separation Induced Transition

With the formulation up to this point, the reattachment downstream of laminar separation occurs much later than should be. To correct this deficiency, the local intermittency is allowed to exceed 1 whenever laminar separation occurs. This results in a large production of which expedites reattachment. The correction for the intermittency function is as follows

96

Turbulence Module

(1)

(2)

(3)

(4)

Turbulence Theory Coupling with Turbulence Model

Although in principle the intermittency function obtained from the model can be applied to any RANS models based on the Boussinesq approximation, the transition model has been calibrated for use with the SST model of Menter. The coupling is as follows

(1)

(2)

where

(3)

(4)

with and the production and destruction terms from the original SST turbulence model and obtained from above. The blending function , responsible for switching between the

and models is modified as follows

97


(5)

(6)

(7)

The modified blending function is used when the transition model is activated.

Transition Models-Empirical Correlations

Transition Models Empirical Correlations Introduction

The quantity in equation 2 of the Transport Equation for Transition Momentum Thickness

Reynolds Number chapter has been determined from a correlation , where

is the local turbulence intensity and is the acceleration as a measure of the pressure gradient. Two alternatives of such correlation are available: Abu-Ghannam and Shaw correlation and Menter correlation.

Turbulence Theory Abu-Ghannam and Shaw Correlation

The empirical correlation of Abu-Ghannam and Shaw is defined as follows::

(1)

(2)

(3)

where

(4)

98

Turbulence Module

Turbulence Theory Menter Correlation

The empirical correlation of Menter is defined as follows:

(1)

(2)

(3)

where-

(4)

(5)

(6)

(7)

(8)

99


(9)

(10)

For numerical robustness the acceleration parameter and the empirical correlation should be limited as follows:

(11)

(12)

(13)

Turbulence Module Constant Turbulent Viscosity

You can specify a constant value for the turbulent viscosity. This method is made available in CFD-ACE+ only for simplicity. It is not recommended for most practical applications.

Turbulence Module User Defined Turbulent Viscosity

CFD-ACE+ enables you to define your own method of calculating turbulent viscosity. User-calculated values of turbulent viscosity are integrated into the solver through the user-subroutine UVISC. The user defined turbulent viscosity option is only implemented for the k- mode and it only sets the turbulent viscosity. When using this option, you do not have to choose the UVISC option under the VC tab.

Turbulence Module Turbulence in Porous Media

Turbulence modeling in porous media is relevant to the calculations of heat and mass transfer, and to the calculation of flow in continuum regions bounding the porous media. Appropriate modifications to the turbulence transport equations for the porous region are unlikely to be available. For k- turbulence models, CFD-ACE+ provides the following simple algebraic functions to estimate the turbulent kinetic energy and turbulent dissipation rate:

(3-53)

and

100

Turbulence Module

(3-54)

Here is local superficial velocity magnitude; is empirical constant of 0.1; I and L are user-supplied average turbulence intensity and length scale in porous regions, respectively. The typical value for I might be 0.01~0.04. L is usually set to be the hydraulic diameter of the porous passage.

Turbulence Module Wall Functions

There are three options for wall functions in turbulence module in V2007:

standard wall function (default)

two-layer model

non-equilibrium model

Non-Equilibrium Model (Pressure gradient sensitize velocity log-law)

This is a two-layer wall function approach, where each wall cell is divided into two layers (see Figure 1)

viscosity dominated sublayer where the shear stress, , is only due to laminar viscosity

fully-turbulent core region where Reynolds stress solely contribute to total shear

These two layers are assumed to be sharply demarcated at yv, a dimensional thickness of the viscous sublayer. It is also assumed that thin-layer assumptions are largely applicable (e.g.

, etc., where x, y, u, and v are local coordinates and mean velocity components in the tangential and normal directions.) The key elements in the non-equilibrium wall functions are:

the Launder and Spalding’s log-law for mean velocity is sensitized to pressure –gradient effects

the two-layer-based concept is adopted to compute the budget of turbulence kinetic energy in the wall neighboring cells

With these assumptions, in the viscous sublayer is given by

(1-1)

In the turbulent core region, employing Boussinesq’s eddy viscosity hypothesis gives:

(1-2)

The shear stress profile in the inner layer can be obtained by integrating the momentum equation in the tangential direction with respect to y:

(1-3)

101


It is a fair approximation that, in the viscous sublayer, the contribution from inertia is negligible. Then we can write for the viscous sublayer:

(1-4)

Figure 1. The two-layer concept

In the fully-turbulent core, the acceleration effect can not be neglected, and experimental evidence indicates that the shear stress is nearly linear in the turbulent core region. Ensuring the shear stress is continuous at y = yv yields a linear profile:

(1-5)

In viscous sublayer:

(1-6)

In the turbulent core region:

(1-7)

where t is given by

(1-8)

The log-law for mean velocity sensitized to pressure gradients is

(1-9)

where

(1-10)

and yv is the physical viscous sublayer thickness, which is computed from

102

Turbulence Module

(1-11)

Two Layer Model

The two-layer approach is used to specify both and the turbulent viscosity in the near wall cells. In this approach, the whole domain subdivided into a viscosity-affected region and a fully-turbulent region. These two regions are determined by the local turbulence Reynolds number, Rey, which is defined as

(1-12)

where y is the normal distance from the wall from the cell centers. In the fully turbulent region (Rey > Re*

y, Re*y = 200), the k- based turbulence model are solved. In the viscosity-affected

near-wall region (Rey Re*y), the one-equation model of Wolfstein [000] is employed. In this

model, the turbulent viscosity, t, is computed from

(1-13)

where l is the length scale and is computed from

(1-14)

The final effective turbulence viscosity is calculated as

(1-15)

The blending factor is defined as

(1-16)

The constant A determines the width of the bending function. By defining a width such that the value of will be within 1% of its far-field value given a variation of Rey (by default, 10% of Re*

y), the result is:

(1-17)

The is computed from

(1-18)

and the length scales lt is computed from Chen and Patel [000]

(1-19)

103


There constants in the length scale formulas are taken from [000]

(1-20)

Turbulence Module Limitations

All the turbulence models in CFD-ACE+ assume isotropy of turbulent transport. The validity of this assumption is questionable for flows with strong streamline curvature, swirling flows, re-circulation and impingement.

Turbulence-Implementation

Turbulence Module Implementation Introduction

The Implementation section describes how to setup a model for simulation using the Turbulence Module. The Implementation section includes:

Grid Generation - Describes the types of grids that are allowed and general gridding guidelines

Model Setup and Solution - Describes the Turbulence Module related inputs to the CFD-ACE-Solver


Turbulence Module Implementation-Grid Generation

The Turbulence Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). Furthermore all grid cell types are supported (quad, tri, hex, tet, prism, poly).

Generally, the grid needs to be clustered near the walls. The dimensionless distance of the adjacent-to-wall cells, y+ value, is a good indication of how fine the grid is near the wall. When choosing any one of the low-Reynolds number models, including the k- model, the y+ value should be below 1.0. When wall functions are used, the y+ value need to be greater than 11.5 for the cell to lie above the viscous sublayer. The recommended y+ range for the high Reynolds number turbulence models is between 30 and 150.

There is no way to estimate the y+ values before running the solution so it is recommended to work from past experience. Graphical output of the y+ values is available for all of the wall boundaries and should be checked to make sure that the y+ values are within acceptable limits. If the calculated y+ is too large, then the grid will need to be refined in those regions.

Grid Parameter R

For Large Eddy Simulations (LES), the grid parameter R is used to measure the resolution of the grid based on length and time scales for turbulent flows (Avva and Sandaram, 1998). For LES, the grid size should lie in the inertial range of scales, beyond the energetic but larger than the dissipation scales. To obtain a suitable grid, first perform a steady RANS simulation using a suitable k-turbulence model and output the R grid parameter:

104

Turbulence Module

where:

le = the energetic scales (k2/3 / )

ld = the dissipation scales (v3 / )1/4

An R value less than zero implies that the resolved scale is larger than the local, energy containing scales, and R>1 implies that the resolved scales are smaller than the active viscous dissipation scales. Therefore, an R grid parameter between 0 and 1 is needed to perform a satisfactory LES calculation.

Model Setup and Solution

Turbulence Module Implementation-Model Setup and Solution-Introduction

CFD-ACE+ provides the inputs required for the Turbulence Module. This section describes the settings specific to the Flow Module. See CFD-ACE+ Overview for general model settings and basic operation. The Implementation section includes:

Problem Type

Model Options

Volume Conditions

Boundary Conditions

Initial Conditions

Solver Control

Output

Turbulence Module Implementation-Model Setup and Solution-Problem Type


Select Turbulence to activate the Turbulence Module and the Flow Module. The Turbulence Module can work in conjunction with any of the other flow related Modules (e.g., Mixing, Cavitation, etc.).

Model Options

Turbulence Module Implementation-Model Setup and Solution-Model Options-Introduction

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. All of the model options for the Turbulence Module are located under the Turbulence tab. Along with specifying the turbulence model, the method for calculating the Wall Functions must also be specified.

105


Model Options Panel - Turbulence Tab

Turbulence Module Implementation-Model Setup and Solution-Model Options-Turbulence Model

Select a turbulence model from the pull-down menu. The options are described in the Turbulence-Theory section and include:

Standard k- Model

RNG k- Model

Kato-Lauder k- Model

Low Reynolds Number k- Model (Chien)

Two-Layer k- Model

k- Model

k- SST Model

Spalart-Allmaras Model

Large Eddy Simulations - SGS Models

o Smagorinsky Model

o Germano's Dynamic Subgrid Scale Model

o Menon's Localized Dynamic Subgrid Scale Model

Constant Turbulent Viscosity

User Defined Turbulent Viscosity

Turbulence in Porous Media

Turbulence Module Implementation-Model Setup and Solution-Model Options-Subgrid Scale (SGS) Models

CFD-ACE+ provides three Subgrid Scale (SGS) models for LES: Smagorinsky, Localized Dynamic, and Dynamic (see SGS Models). To activate a model, select one from the SGS Model pull down menu.

106

Turbulence Module

Turbulence Tab - Large Eddy Simulation - SGS Model

The Smagorinsky SGS model requires two additional parameters: the model constant Cs0, and the Vandriest Damping. The default for these constants are reasonable values for typical LES applications.

In order to activate one of the subgrid scale models (LES), the time dependence should be set to Transient on the MO Shared tab.

Turbulence Module Implementation-Model Setup and Solution-Model Options-Turbulent Prandtl Number

If you activate the Heat Transfer Module, you can specify a turbulent Prandtl number. This models the effect of turbulence on heat transfer through an effective conductivity:

(3-55)

Experiments have generally shown the value of t to range from about 1.0 near walls to values of 0.7 or less as the distance from walls increases. The default value of 0.9 is a reasonable compromise between these bounds.

Turbulence Module Implementation-Model Setup and Solution-Model Options-Turbulent Schmidt Number

When you activate the Chemistry Module, you can specify a turbulent Schmidt number. This models the effect of turbulence on mass diffusion through an effective diffusivity:

(3-56)

Turbulence Module Implementation-Model Setup and Solution-Volume Conditions

No volume condition inputs are required for the Turbulence Module.

Boundary Conditions

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Introduction

107


Click the Boundary Conditions [BC] tab to see the Boundary Conditions Panel. See Control Panel-Boundary Condition Type for details. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or the BC Explorer.

The Turbulence Module is fully supported by the Cyclic, Thin Wall, and Arbitrary Interface boundary conditions. See Cyclic Boundary Conditions, Thin-Wall Boundary Conditions or Arbitrary Interface Boundary Conditions for details.

All of the general boundary conditions for the Turbulence Module are located under the Turbulence tab and can be reached when the boundary condition setting mode is set to General. Each boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). The Turbulence Module Boundary Condition section includes:

Boundary Conditions-Inlets

Boundary Conditions-Outlets

Boundary Conditions-Turbulent Kinetic Energy

Boundary Conditions-Random Inlets

Boundary Conditions-Walls

Boundary Conditions-Rotating Walls

Boundary Conditions-Symmetry

Boundary Conditions-Interfaces

Boundary Conditions-Thin Walls

Boundary Conditions-Cyclic

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Inlets

The Turbulence Module needs to know how to set the turbulence quantities at inlet boundaries.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Outlets

You can set values for turbulent kinetic energy (K) and turbulence dissipation rate (D) at flow outlets. These values will only be used where there is inflow through the outlet boundary). The turbulence quantities that need to be specified for the turbulence models are:

Turbulence Quantity Model

Turbulent Kinetic Energy (K) k- or k-Models

Turbulent Dissipation Rate (D) k- or k- Models

Eddy Viscosity (Nu(t)) Spalart-Allmaras Model

SGS Turbulent Kinetic Energy (Ksgs) localized dynamic SGS

Turbulent Kinetic Energy

108

Turbulence Module

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Turbulent Kinetic Energy

Turbulent kinetic energy can be specified as:

Constant

Turbulence intensity (0~1)

Profile X (input (x, k) data pairs)

Profile Y (input (y, k) data pairs)

Profile Z (input (z, k) data pairs)

Profile 2D (input (x, y, z, k) data sets)

Profile in time (input (t, k) data pairs)

Profile from file (see user manual A-4 Profile BC file)

Parametric (define a function from parametric input panel)

User subroutine (see User Manual Chap. 11 for details)

Turbulent dissipation rate/Specific dissipation rate can be specified as:

Constant

Length scale

Profile X (input (x, D) data pairs)

Profile Y (input (y, D) data pairs)

Profile Z (input (z, D) data pairs)

Profile 2D (input (x, y, z, D) data sets)

Profile in time (input (t, D) data pairs)

Profile from file (see user manual A-4 Profile BC file)

Parametric (define a function from parametric input panel)

Hydraulic diameter

User subroutine (see User Manual Chap. 11 for details)

Eddy viscosity can be specified as:

Constant

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Turbulence Quantities Using Turbulence Intensity

The turbulence intensity, I, is defined as the ratio of the root-mean-square of the fluctuation

velocity, , to the mean flow velocity,

The turbulence intensity generally ranges from 1% to 10%. That with turbulence intensity less than 1% is considered as low turbulent flow and that with turbulence intensities greater than 10% are considered as high turbulent flows.

The turbulence intensity at the core of a fully developed duct flow can be estimated as:

109


where is the Reynolds number.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Estimating Turbulent Kinetic Energy from Turbulence Intensity

For boundaries and volumes (initialization) with turbulence intensity as the input option, the turbulent kinetic energy can be estimated from:

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Turbulence Length, Scale, and Hydraulic Diameter

The turbulence length scale, , is a physical quantity related to the size of the large eddies that contain the energy in turbulent flows.

In fully developed pipe or duct flow, is restricted by the size of the duct. A relationship between

and the hydraulic diameter L is:

= 0.03L

Guidelines for choosing hydraulic diameter L or turbulence scale :

(1) For fully developed internal flows, choose hydraulic diameter method and input the characteristic length of the flow in/outs as hydraulic diameter.

(2) For wall-bounded flows in which the inlets involve boundary layer, choose turbulence length

scale. Set =0.4 , is the thickness of boundary layer.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Turbulent Dissipation Rate

For boundaries and volumes (initialization) with turbulence length scale or hydraulic diameter as input option, the dissipation rate, (or the specific rate of dissipation, ) can be calculated as:

k- model

k- model

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Random Inlets

Under the BC/Turb Tab, select the Random Inlet check box to specify a Gaussian or time correlated randomization of the inlet velocity components.

110

Turbulence Module

Boundary Condition - Turbulence Tab - Gaussian Random Option

Both the Gaussian and time correlated options randomly perturb the mean inlet velocity components over the surface of the inlet boundary using a Gaussian profile of the root mean squared (RMS) turbulent intensities at each time step. The time correlated option further specifies that the perturbation is correlated over a length of time.

Turbulence Module Walls

Wall-Roughness boundary condition has been implemented in the ACE+ code. The new feature allows one to account for sand-grain roughness when the standard wall-function approximation is used.

Formulation

111


Roughness will increase the drag over that on a hydraulically smooth surface. This increase is

reflected in the downward shift of the velocity profile presented in coordinates, as is demonstrated in Schlichting's Boundary Layer Theory. Therefore, the logarithmic law for velocity distribution

(1)

which forms the basis of the Wall-Function approach is no longer valid in the presence of

roughness. However, presented in , rather than , the log-law is still valid in form for complete sand roughness,

(2)

here is the roughness height and is an empirical constant of 30.0. The above equation has been widely used as the basis for roughness wall-function.

In order to generalize our standard wall-function approach in ACE+, equation 2 has been recast into the following form:

(3)

Clearly, when the roughness equation reverts to the one for a smooth

surface (equation 1). At this level of , the size of the roughness is so small that all protrusions are contained within the laminar sub-layer. The surface is regarded as hydraulically smooth. (In

fact, Nikuradse's experiments show that when is less than 5, roughened pipes have the same resistance as smooth pipe.) In implementing equation 3 in the ACE+ code, the effect of

roughness comes into play by setting the coefficient to in the existing wall-function

method. In so doing, is evaluated through equation 2, and it is set to once it falls below this value for reasons discussed above.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Rotating Walls

Rotating walls, just like plain walls, can have a roughness height (RH) value assigned.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Symmetry

The symmetry boundary condition is a zero-gradient condition. There are no Turbulence Module related values for symmetry boundary conditions.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Interfaces

112

Turbulence Module

The interface boundary condition is used to allow two computational domains to communicate information. There are no Turbulence Module related values for interface boundary conditions.

Interface boundary conditions can be converted to thin walls. See Thin-Wall Boundary Conditions and Arbitrary Interface Boundary Conditions for details on other ways for computational domains to communicate.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Thin Walls

The Thin Wall boundary condition is fully supported by the Turbulence Module. See Thin-Wall Boundary Conditions for instructions on how to setup a thin wall boundary condition. The Turbulence Module treats a thin wall boundary condition the same as a Wall boundary condition. See Boundary Conditions-Walls.

Under the Turbulence tab, there are inputs available for roughness height specification. This roughness height will be applied to both sides of the thin wall boundary condition.

Turbulence Module Implementation-Model Setup and Solution-Boundary Conditions-Cyclic

The Cyclic boundary condition is fully supported by the Turbulence Module. See Cyclic Boundary Conditions for instructions on how to setup a Cyclic boundary condition. There are no Turbulence Module related settings for the Cyclic boundary condition.

Turbulence Module Implementation-Model Setup and Solution-Initial Conditions


The Initial Conditions can either be specified as constant values or read from a previously run solution file. If constant values are specified, you must provide initial turbulence values. The values can be found under the Turbulence (Turb) tab and the following variables must be set:

Turbulent Kinetic Energy (K)

Turbulent Dissipation Rate (D)

Eddy Viscosity (Nu(t)) for the Spalart Allmaras Model

RMS u', v', w' turbulent intensities for random initial conditions and LES

If a previous solution is used for restart and a random perturbation is desired, select the restart from RANS checkbox. This will use the kinetic energy from the RANS calculation to perturb the velocity field.

Although, for a steady state problem, the Initial Condition values do not affect the final solution, reasonable values should be specified so that the solution does not have convergence problems at start-up.

Because the turbulence values produce an effective viscosity, and increased viscosity can make the solution more stable, sometimes it is useful to set somewhat larger values of K (or smaller values of D) to increase the initial effective viscosity field.

Model Setup and Solution-Solver Control

Turbulence Module Implementation-Model Setup and Solution-Solver Control-Introduction

113


Click the Solver Control [SC] tab to see the Solver Control Panel. See Control Panel-Solver Controls for details.

The Solver Control page allows access to the various settings that control the numerical aspects of the CFD-ACE-Solver as well as all of the output options. The Solver Control section includes

Solver Control-Output for LES

Solver Control-Spatial Differencing Scheme

Solver Control-Solver Selection

Solver Control-Under Relaxation Parameters

Solver Control-Variable Limits

Solver Control-Advanced Settings

Turbulence Module Implementation-Model Setup and Solution-Solver Control-Output for LES

Running Averages of the flowfield variables can be computed by setting the Start Timestep in the Large Eddy Simulation (Averaging) output panel.

The Save LES Statistics option saves the running average variables for restart purposes. Only the variables that are checked for Graphical Output will be saved. For continuation of the averaging process from restart data the corresponding option must be checked under the Previous solution menu from IC Sources.

Turbulence Module Implementation-Model Setup and Solution-Solver Control-Spatial Differencing Scheme

Under the Spatial Differencing tab, select the differencing method to be used for the convective terms in the equations. Activating the Turbulence Module enables you to set turbulence equations. The default method is first order Upwind. We recommend to always use the first order Upwind method for the turbulence equations as the higher order schemes can produce convergence problems and do not increase the solution accuracy significantly. See Spatial Differencing Scheme for more information on the different differencing schemes available. See Numerical Methods-Central Differencing Schemes for numerical details of the differencing schemes.

Turbulence Module Implementation-Model Setup and Solution-Solver Control-Solver Selection

Under the Solvers tab, select the linear equation solver to be used for each set of equations. Activating the Turbulence Module enables you to set turbulence equations. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 50 sweeps. The default convergence criteria is 0.0001. See Solver Selection for more information on the different linear equation solvers available. See Linear Equation Solvers for numerical details of the linear equation solvers.

Turbulence Module Under Relaxation Parameters

Under the Relaxation tab, select the amount of under-relaxation to be applied for each of the dependent (solved) and auxiliary variables used for the flow equations. Activating the Turbulence Module enables you to set turbulence variables, as well as the auxiliary variable, viscosity. See Under Relaxation Parameters for details on the mechanics of setting the under relaxation values.

114

Turbulence Module

See Numerical Methods-Under Relaxation for numerical details of how under-relaxation is applied.

The turbulence equations use an inertial under relaxation scheme and the default values are 0.2. Increasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.

The calculation of viscosity uses a linear under relaxation scheme and the default values are 1.0. Decreasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.


Turbulence Startup Control

Turbulent flow simulations can sometimes exhibit diverging behavior at the beginning of a calculation. The Turbulent Start Control feature provides a method of constraining the change of turbulent viscosity at the start of a simulation with the aim of eliminating the divergence. Inputs for the control appear when the check box is selected and are shown below

Turbulence Start Control Inputs

The initial turbulent viscosity that will be used will be calculated from the Viscosity Ratio input and will be equal to the Viscosity ratio times the molecular viscosity. This is contrasted to the normal calculation of the initial turbulent viscosity from the initial values of the turbulence quantities. The default value of 1000 will usually have a reasonably stabilizing effect on the calculations.

The Initial Iterations input is the number of iterations after startup for which the viscosity will be held constant at the initial value.

The Transition Iterations input is the number of iterations over which to linearly transition from an unchanging turbulent viscosity field (linear under-relaxation of 0.0) to a viscosity field under-relaxed at the previously specified value of linear under-relaxation.

Turbulence Module Implementation-Model Setup and Solution-Solver Control-Variable Limits

Under the Limits tab, set the minimum and maximum allowed variable values. CFD-ACE+ will ensure that the value of any given variable will always remain within these limits by clamping the value. Activating the Turbulence Module enables you to set limits for K, D, and Viscosity variables. See Variable Limits for details on how limits are applied.

The default limits should be used. For the Turbulence Module however, it has been found that applying a maximum limit on viscosity can sometimes help to get through some convergence problems. Ensure that you check the solution to verify that the final solution is not constrained by the imposed limit (which could produce un-physical results).

Turbulence Module Implementation-Model Setup and Solution-Solver Control-Advanced Settings

115


In CFD-ACE+, by default, inertial under-relaxation of dependent variables is used to constrain the change in the variable from one iteration to the next in order to prevent divergence of the solution procedure.

The default inertial relaxation method can be switched to the CFL based relaxation method by going to SC-->Adv and checking the appropriate check boxes for each module.


The relaxation factor defined in SC-->Relax is used as the CFL multiplier. A general rule of thumb would be the inverse value of usual inertial relaxation factor.

Effect of Value:

5 = Default Value



Model Setup and Solution-Output

Turbulence Module Implementation-Model Setup and Solution-Output-Introduction

Click the Out tab to see the Output settings in the Control Panel. The output section includes:

Output-Printed Output

Output-Graphical Output

Output-LES Output

Turbulence Module Implementation-Model Setup and Solution-Output-Printed Output

There are no printed output summaries available for the Turbulence Module.

Turbulence Module Implementation-Model Setup and Solution-Output-Graphical Output

Under the Graphics tab, you can select the variables to output to the graphics file (modelname.DTF). These variables will then be available for visualization and analysis in CFD-VIEW. Activation of the Turbulence Module allows output of the variables listed:

Turbulence Module Related Graphical Output

Variable Units

Turbulent Kinetic Energy m2/s2

Turbulent Dissipation Rate m2/s3

Turbulent Viscosity kg/m-s

116

Turbulence Module

Eddy Viscosity m2/s

Effective Viscosity (sum of turbulent and laminar viscosity)

kg/m-s

Y+ (only output at walls) -

Turbulence Module Implementation-Model Setup and Solution-Output-LES Output

The output variables available for LES are listed in the table.

LES Module Graphical Output

Variable Units Model*

Turbulent Intensities m/s S,D,LD

Y+ - S,D,LD

Eddy Viscosity m2/s S,D,LD

Strain Invariant 1/s S,D,LD

Vorticity 1/s S,D,LD

Dynamic Coefficient - D

Ctau - LD

Ceps - LD

SGS Kinetic Energy m2/s2 LD

SGS Dissipation Rate m2/s3 LD

Test Filter Kinetic Energy m2/s2 LD

117


m/s S,D,LD

* S - Smagorinsky D - Dynamic LD - Localized Dynamic

Turbulence Module Implementation-Post Processing

CFD-VIEW can post-process the turbulence solutions. Two important quantities that need to be looked at in the graphical output are the level of turbulent kinetic energy (or turbulent viscosity) and y+. Turbulence levels are high in regions where the rate of strain is high, such as near-wall regions and regions of flow re-circulation and stagnation. The values of y+ at the walls is good indication of the level of grid refinement near the wall. A complete list of post processing variables available as a result of using the Turbulence Module is shown in the table below.


D Dissipation Rate(k model)

Specific rate of dissipation (k model)

m2/s3

s-1

K Kinetic energy m2/s2

ED_VIS Eddy Viscosity (Spalart-Allmaras model) m2/s

VIS_T Turbulent Viscosity kg/m-s

YPLUS Yplus values -

Turbulence Module Frequently Asked Questions

Which turbulence model should I choose?

This really depends upon your need. If you just want to consider the overall effect of turbulence on the mean flow field, rather than some fine details, you may choose one of the high Reynolds number models that are more robust and cost-effective. In this case, the standard k- model can be chosen for most problems. However, when there is strong flow recirculation the RNG model is a better choice; and for cases with strong flow stagnation the Kato-Launder model becomes a better choice. On the other hand, If your are interested in fine details such as heat transfer coefficients or viscous wall friction, you should choose one of the low Reynolds number turbulence models.

How do I specify initial turbulent quantities?

For steady-state simulations, the initial conditions will not affect the final solutions. But they may affect numerical stability. It has been found that a low level of turbulence intensity helps convergence. Generally, you may set turbulent kinetic energy, k, to be one percent of the initial or inlet mean kinetic energy. Then you may specify a value for with which the calculated turbulent

118

Turbulence Module

viscosity is about 20 times the laminar viscosity. For transient calculations, since the initial conditions will affect the final results, ideally you should specify values for turbulence quantities based on experimental data whenever they are available. If they are not available, you may follow the above suggestions for steady-state simulations.

How do I specify turbulence quantities at inlet boundaries?

For simulations or regions where convective transport is considerably greater than turbulence production (usually occur in the absence of strong mean flow velocity gradients) it is the inlet conditions of turbulent quantities that determine the overall level of turbulent viscosity. Again, you may follow instructions as given for the initial conditions.

How do I specify turbulent quantities at outlet boundaries?

When flow goes out at the outlet, zero-gradient boundary conditions are used for the turbulence quantities. Only when flow comes back into the computational domain are the boundary values of turbulence quantities used. Specification of boundary values may also follow the above suggestions for the inlet BCs.

Turbulence Module Examples

The following tutorials use the Turbulence and Flow Modules exclusively:

Tutorial 2, Turbulent Flow Past a Backward Facing Step in Tutorial Manual, Volume II.

The following tutorials use the Turbulence and Flow Modules in conjunction with one or more other Modules:

Tutorial 7, Turbulent Mixing of Propane and Air in Tutorial Manual, Volume II, (with and without reactions).

Turbulence Module References

Avva, R.K., and Sundaram, S., “Numerical Simulation of Surface Pressure Fluctuations in Complex Geometries." CFDRC SBIR Phase II Final Report, Navy Contract N000114-98-CO416, CFDRC Report No. 480316, 1995.

Chen, H. C., and V. C. Patel., “Near-Wall Turbulence Models for Complex Flows Including Separation." AIAA Journal 26.6 (1988): 41-648.

Chien K.Y., "Prediction of Channel and Boundary-Layer Flows with a Low Reynolds Number Turbulence Model." AIAA Journal 20.1(1982): 33-38.

Ciofalo, M., and Collins, M.W., “k-Predictions of Heat Transfer in Turbulent Recirculating Flows Using an Improved Wall Treatment.” Numer. Heat Transfer 15(1989): 21-47.

Germano, M., (1992), "Turbulence: The Filtering Approach." J. Fluid Mechanics 238, pp. 325-336.

Givi, P., (1989), "Model Free Simulations of Turbulent Reactive Flows." Prog. Energy Combust. Sci., 15, pp. 1-107.

Hellsten, A., “Extension of the k--SST turbulence model for flows over rough surface.” AIAA-97-3577.

Kim, W., and Menon, S., (1997), "Application of the Localized Dynamic Subgrid Scale Model to Turbulent Wall-Bounded Flows." AIAA paper 97-0210.

Launder, B.E., and Spaulding, D.B., “The Numerical Computation of Turbulent Flows.” Comp. Methods for Appl. Mech. Eng. 3(1974): 269-289.

119


120

Lilly, D.K., (1992), "A Proposed Modification of the Germano Subgrid Scale Closure Method." Phys.Fluids 4, pp. 633-634.

Menter., F.R., “Zonal two equation k- turbulence models for aerodynamic flows." AIAA-93-2906.

Smagorinsky, J., (1963), "General Circulation Experiments with the Primitive Equations, I. The Basic Experiment." Monthly Weather Review 91, pp. 99-96.

Spalart, P.R., and Allmaras, S.R., “A One-Equation Model for Aerodynamic Flows.” AIAA Journal 92:439.

Wilcox, David C., Turbulence Modeling for CFD. La Canada, California: DCW Industries, 1993.

Yakhot, V., Orszag, S.A., Thangam, S., Speziale, C.G., Gatski, T.B., “Development of Turbulence Models for Shear Flows by a Double Expansion Technique.” Phys. Fluids A 4.7 (1992): 1510-1520.

Yakhot, V., and Orszag, S.A., “Renormalization Group Analysis of Turbulence.” J. Sci. Compute. 1.1(1986) 3-51.

Chemistry Module

Chemistry Module Introduction

The Chemistry Module enables you to solve mixing and reacting flow problems. Activating the Chemistry Module implies the solution of the mixture or species mass fractions, (the latter requiring solution of additional mass transport equations). If you activate liquid chemistry, instead of solving transport equations for mass fractions, transport equations for molar concentration are solved. You can use the Chemistry module to study systems where both surface and gas-phase reactions occur. Reactions involving charged species (encountered in plasma reactors) can also be studied. You can also use it to study electrochemistry problems such as fuel cells of those involving charged particle species transport in the liquid phase. (See Applications: Electrochemistry for details.) The Chemistry Module includes:

Chemistry-Applications

Chemistry-Features

Chemistry-Theory

Chemistry-Turbulence-Combustion Interaction

Chemistry-Limitations

Chemistry-Implementation

Chemistry-Frequently Asked Questions

Chemistry-Examples

Chemistry-References

Chemistry-Applications

Chemistry Module Applications-Introduction

Mixing and reacting flows are encountered in a wide variety of applications such as combustors, chemical and plasma reactors, and gas-turbines. A detailed model of the velocity and temperature field and species concentrations can greatly aid the design, optimization, and control of these systems. You can use the Chemistry Module:

To study processes such as deposition and etching that are vital in semiconductor processing applications.

For mixing-only cases and gas-phase and/or surface reactions prescribed within the volumes and/or at surfaces.

With other CFD-ACE+ modules to study multi-physics problems.

The Chemistry Applications section includes:

Applications-Mixing Only

Applications-Mixing with Gas Phase Reactions

Applications-Mixing with Surface Reactions

Applications-Multi-Physics Applications

121


Chemistry Module Applications-Mixing Only

The Chemistry Module simulates mixing two or more inert species or mixtures. The spatial and temporal variation of the species concentrations can be obtained using the Chemistry Module.

Chemistry Module Applications-Mixing with Gas Phase Reactions

You can use the Chemistry Module to study systems involving chemical reactions. Two examples are combustion problems and semiconductor process chamber simulations. A transport equation is solved for each mixture or species with a source term representing the net rate of production or depletion of the mixture or species. The reactions can take the form of instantaneous, equilibrium, or finite-rate mechanisms. For combustion problems, several reduced mechanisms are available.

For combustion problems several reduced mechanisms are available. See the Database Manager.

Chemistry Module Applications-Mixing with Surface Reactions

Use the Chemistry Module to model surface reactions occurring in chemical vapor deposition (CVD) systems.

Chemistry Module Applications-Multi-Physics Applications

The Chemistry Module is often used with (and is required by) many of the other modules in CFD-ACE+ to perform multi-physics analyses. Some of the more commonly added modules are listed below. Examples of these types of applications are given in each Module’s chapter.

Flow

Turbulence

Heat Transfer (with or without radiation)

User Defined Scalars

Spray

Plasma

VOF

Electric

Chemistry-Features

Chemistry Module Features Introduction

The Chemistry Module has many inherent features which may or may not be activated for any given simulation. The Features section includes:

Features-Solution Approach

Features-Mass Diffusion Options

Features-Gas Phase Reactions

Features-Surface Reactions

Features-Coupled Solver

122

Chemistry Module

Features-Unsteady Combustion

Chemistry Module Features-Solution Approach

The Chemistry module has two solution approach options: Mixture Mass Fractions and Species Mass Fractions. Each approach has its advantages and disadvantages and they are briefly described below.

Mixture Mass Fractions

The Mixture Mass Fraction approach requires a solution of fewer transport equations than the Species Mass Fraction option. However, some models and fluid property options are not available for the Mixture Mass Fraction approach. The Mixture Mass Fraction approach is usually used for pure mixing problems and combustion reaction problems involving reactions which are either in equilibrium, very fast (instantaneous), or can be modeled with a single global finite-rate reaction step. Diffusivity of individual species is not accounted for with this option since all mixtures are considered to have the same valve. The mixture mass fraction approach can be used to model turbulence/chemistry interaction through either eddy-breakup or assumed pdf methods. Models for CO oxidation and NOx production are also available.

Species Mass Fractions

The Species Mass Fraction approach is the most general approach and encompasses all problems that can be solved using the Mixture Fraction approach except for models that include turbulence/chemistry interaction. The Species Mass Fraction approach requires the solution of a transport equation for every species in the system. This approach is required for:

Multi-component diffusion problems

Surface reaction problems

A multi-step finite rate gas-phase reaction

Chemistry Module Features-Mass Diffusion Options

There are two options available for mass diffusion: constant Schmidt number and multi-component diffusion. The multi-component diffusion model is only available when the Species Mass Fraction solution approach has been selected.

Species Conservation Options: When species diffuse at different rates, their mass fractions do not automatically add up to unity, and some corrections have to be invoked to guarantee species conservation. The following options are available:

None: no corrections are invoked, and species mass fractions may not add up to unity. This option is equivalent to not invoking conservation at all.

Reference Specie: If mass fractions do not add up to unity, the mass fraction of the reference species is adjusted to enforce conservation.

Stefan-Maxwell: Species conservation is enforced by employing the Stefan-Maxwell equations. This is the most rigorous of all the approaches, but is computationally more expensive.

Features-Gas Phase Reactions

Chemistry Module Features-Gas Phase Reactions-Introduction

The Chemistry Module contains the following gas phase reaction models:

Instantaneous Reaction Model (for Mixture Mass Fraction approach)

123


Equilibrium Reaction Model (for Mixture Mass Fraction approach)

Finite-Rate Model (for Mixture Mass Fraction approach)

Finite-Rate Model (for Species Mass Fraction approach)

Eddy Breakup Model

Prescribed PDF Model

All of these gas phase reaction mechanisms are setup using the Reaction Manager. See the Database Manager for details.

Chemistry Module Features-Gas Phase Reactions-Instantaneous Reaction Model

The Instantaneous Reaction Model assumes that a single chemical reaction occurs and that it proceeds instantaneously to completion. You can only use this model if the Mixture Mass Fraction solution approach has been selected. The mixture fraction assumed PDF model may be used with an instantaneous reaction.

Chemistry Module Features-Gas Phase Reactions-Equilibrium Reaction Model

The Equilibrium Reaction Model (Pratt and Wormeck, 1976) assumes that chemical reactions are so fast that the mixture is in chemical equilibrium. The main difference between this model and the instantaneous model is that the user does not have to specify a stoichiometrically balanced reaction. The composition (stoichiometry) is determined by minimizing the Gibbs energy of the system. You can only use this model if the Mixture Mass Fraction solution approach has been selected.

Chemistry Module Features-Gas Phase Reactions-Finite-Rate Model (for Mixture Solution)

The Finite-Rate Model (for mixture mass fraction approach) enables you to specify a single reaction step which proceeds at a finite-rate. This model is restricted to two reactant species. The primary difference between this finite-rate model and the instantaneous model is that the mass fraction of fuel is calculated by solution of a transport equation with a source term due to chemical reaction for the finite-rate model. The mass fractions of the other species are calculated from the mixture fractions and the mass fraction of fuel. This model can only be used if the Mixture Mass Fraction solution approach has been selected. Turbulence/chemistry interaction can be accounted for using either the eddy breakup or assumed PDF models discussed below.

If a multi-step reaction is desired then the Species Mass Fraction approach must be used and hence the Finite-Rate Model for Species Solution is appropriate.

Chemistry Module Features-Gas Phase Reactions-Finite-Rate Model (for Species Solution)

The Finite-Rate Model (for Species Mass Fraction approach) enables you to specify any number of reaction steps which each proceed at a finite-rate. This model does not have any restrictions on the number of reactant species and third-body effects can also be included. For plasma reactions, an electron-induced reactions can be specified. This model can only be used if the Species Mass Fraction solution approach has been selected.

Two options are available to specify the type of finite rate reactions. If the Mass Fraction option is selected, the law of mass action is used to compute the reaction rates. The backward rate (if specified) is calculated by assuming equilibrium. The reactant and product exponents are equal to their stoichiometric coefficients. If you select the General Rate option, the law of mass action is not used, and the reactant and product exponents can be arbitrary. If you specify backward

124

Chemistry Module

reaction, the backward reaction rate can be computed using prescribed values, or by using equilibrium.

Chemistry Module Features-Gas Phase Reactions-Eddy Breakup Model

You can use the eddy breakup model for turbulence-combustion interaction for turbulent flows, with any of the k- turbulence models, and with the mass fraction finite-rate reaction model. This model limits the reaction rate where turbulent mixing controls the mixing of segregated reactant species or of premixed reactants and hot products.

Chemistry Module Features-Gas Phase Reactions-Prescribed PDF Model

The equations solved by CFD-ACE+ for turbulent reacting flows are transport equations for density-weighted mean values. However, auxiliary variables such as density and temperature are really nonlinear functions of the composition. These variables can be calculated more accurately by integrating the product of the variable of interest and the density-weighted joint composition probability density function (PDF) over the range of composition values. The source terms for finite-rate reactions are highly nonlinear and should be calculated similarly. The shape of a PDF for the mixture fraction can be prescribed (assumed) in CFD-ACE+ to model turbulent combustion when separate fuel and air (oxidizer) mixtures are defined.

Chemistry Module Features-Surface Reactions

The surface reaction models allow the calculation of deposition, etching, or catalytic reactions at surfaces and hence can model systems where these processes are of importance.

All surface reactions can be specified using a multi-step finite-rate reaction mechanism. The reaction rates of individual steps can be computed either by using the sticking coefficient model, or by using a general finite-rate expression. In CFD-ACE+, steps involving these two approaches can be mixed. Reaction mechanisms involving surface-adsorbed species and site coverages can be modeled using this feature. For problems involving plasma (i.e., when the plasma module is turned on), it is also possible to model neutralization of charged species on the walls, in conjunction with regular neutral species reactions.

All of the surface reaction mechanisms are setup using the Surface Reaction Manager. See Database Manager-Surface Reaction for details.

Chemistry Module Features-Coupled Solver

In multi-step finite rate reactions, it is possible that one of the reaction steps proceeds at a rate that is orders of magnitude higher than the other reactions. The numerical solution of the system of equations describing the time-evolution of the various species is fraught with difficulties. A system of differential equations with widely varying time constants is called stiff. The Chemistry module can handle both stiff and non-stiff systems. Select the Coupled Solver option if the reaction set under consideration has some fast transients.

When the Coupled Solver is turned on, the transport equations for all the species are solved in a coupled manner, rather than in a segregated manner. The convergence is generally slower but more stable. It is suitable for all types of chemistry, not just surface chemistry. There is no relaxation associated with the coupled solver.

Chemistry Module Features-Unsteady Combustion

For unsteady reactive flow simulations a few methodologies specific to combustion problems are available to either accelerate the calculations or to increase the accuracy level of the results. In

125


the case of complex reaction mechanisms the Laminar Chemistry Operator Splitting option allows for the usage of the In Situ Adaptive Tabulation (ISAT) method and/or the Staggered Chemistry solution approach to considerably expedite the numerical simulation.

For Large Eddy Simulation, an accurate subgrid chemistry closure (the Linear Eddy Model) is available along with the ISAT and Staggered Chemistry options.

Chemistry-Theory

Chemistry Module Theory Introduction

The Chemistry Module enables you to model mixing and reacting flow systems. The Chemistry Module Theory section includes:

Theory-Basic Definitions And Relations

Theory-Gas Phase Reaction Models

Theory-Surface Reaction Models

Chemistry-Theory-Definitions And Relations

Chemistry Module Theory-Definitions and Relations-Introduction

Calculation of reactive flow requires the consideration of both stoichiometry and reaction kinetics. Stoichiometry is the description of the conservation of mass and elements. Reaction kinetics is the description of the individual steps that make up a chemically reacting system and the specification of the rates at which those steps progress.

A distinction will be made between elementary and global reactions. A global reaction is one such as:

CH4 + 2 O2 CO2 + 2 H2O (4-1)

which is correct in the stoichiometric sense, because all elements are conserved. This global step does not describe the true path of methane combustion, which is made up of many elementary reaction steps:

CH4 + H CH3+ H2 (4-2)

Elementary reactions describe the intermediate steps in a chemical reaction, which are representative of actual collisions between molecules.

The Definitions and Relations section includes:

Theory-Definitions and Relations-Composition Variables

Theory-Definitions and Relations-Chemical Rate Expressions

Theory-Definitions and Relations-Mixture Fractions

Chemistry Module Theory-Definitions and Relations-Composition Variables

Several different composition variables are used for flow with mixing or reaction. The mass fraction of species i in a multi-component system, Yi, is defined as the mass of the ith species per unit mass of the mixture. Similarly, the mole fraction xi is defined as the number of moles of the ith species per mole of the mixture. The mole and mass fractions are related to each other by the molecular weight of the ith species, Mi, and the mixture molecular weight, M.

126

Chemistry Module

(4-3)

The mixture molecular weight is given by:

(4-4)

The molar concentration of species i, ci, is defined as the number of moles of the ith species per unit volume. It is related to Yi as:

(4-5)

where , the mixture density, is computed from the equation of state.

The number density (#/m-3) of a species i, is obtained by multiplying the molar concentration with Avogadro’s number (6.023x1023 1/mol). The number of moles of species i per unit mass, ni, is defined as:

(4-6)

and is a useful quantity in converting concentration units, as can be seen by examining equations 4-3 through 4-5. The partial pressure of species i in a mixture of gases is defined as:

(4-7)

Chemistry Module Theory-Definitions and Relations-Chemical Rate Expressions

A system of Nrxn chemical reactions involving Nsp species can be expressed in a general notation by:

(4-8)

where i is the chemical symbol for species i, v'ij and v''ij are the forward and reverse stoichiometric coefficients for the ith species in the jth reaction. Equation 4-8 can be written more compactly as:

127


(4-9)

where vij= v''i -v'ij. The chemical reaction must be balanced (i.e., the total number of atoms of each element must be the same on both sides of equation 4-8). The stoichiometric coefficients are integers for elementary reactions and are normally 0, 1, or 2. Elementary reactions usually involve no more than four species, so the array of stoichiometric coefficients is sparse.

The nomenclature given above is illustrated in the following example. A system containing the species H2, H2O, CO, CO2, O2, and N2 may have the following reactions:

CO + H2O = CO2 + H2

2 H2 + O2 = 2 H2O

For this system the stoichiometric coefficients for the above reactions are:

The molar production rate of species i due to chemical reaction is

(4-10)

The rate-of-progress variable for the jth reaction, qj, can be generally expressed as:

(4-11)

128

Chemistry Module

where:

and

are temperature-dependent forward and reverse rate coefficients

and

are constants

For elementary reactions which obey the mass action law:

=

and

=

,

where:

and

are the stoichiometric coefficients defined in equation 4-8.

The concentration exponents in equation 4-11 are not necessarily related to the stoichiometric coefficients for global reactions.

The rate coefficients are assumed to have an Arrhenius form:

(4-12)

where:

A = pre-exponential constant

n = temperature exponent

Ea/R = activation temperature

m = exponent on pressure dependency

where A, n and Ea/R are constants for each reaction. (The subscript j has been deleted for clarity.) The units of the reaction rate given by equation 4-11 are (moles/volume/time). The units of A, therefore, depend on the exponents of the molar concentrations in equation 4-11. Note that units for concentration reported in the literature are typically g-moles/cm3, while the units used in CFD-ACE+ are kg-moles/m3. In other words, for a simple reaction of the form:

(4-13)

the rate of the reaction is expressed as:

(4-14)

129


where:

is expressed in kmoles/m3s

The units on Ap are dependent upon ,, and n as shown below:

units of Ap

=

(4-15)

The reverse rate coefficient can be obtained from the equilibrium constant, Kc, for reactions obeying the law of mass action:

(4-16)

The equilibrium constant (actually a function of temperature) can be calculated from thermodynamic data:

(4-17)

where:

p0 = reference pressure of one atmosphere

=

Gibbs free energy of species i at one atmosphere.

Elementary reactions are sometimes written with a third body, usually designated with the symbol M, and can be any species. For example:

H + O2 + M = HO2 + M

The rate-of-progress variable for these reactions is:

(4-18)

\where:

130

Chemistry Module

= Third body efficiency

= Order for third-body

Chemistry Module Theory-Definitions and Relations-Mixture Fractions

Flows with mixing or reaction can be calculated by solving transport equations for the mass fraction of all (or all but one) species. The number of variables needed to calculate the flow can be reduced, in certain cases, by introducing variables referred to as mixture fractions. A mixture is defined as a combination of species with a fixed composition. For example, a mixture designated air may have a composition of 23.2% O2 and 76.8% N2 by mass whereas a mixture designated fuel may have a composition of 100% CH4.

Each mixture in CFD-ACE+ is tracked with a mixture fraction variable, which is governed by the general transport equation

(4-19)

In the preceding equation fk represents the mixture fraction for the kth mixture. Note that this equation contains no source terms due to chemical reaction. The only source term is due to the evaporation of spray droplets. The diffusion coefficient () is the same for all mixture fractions.

Mixture fractions are normally associated with one or more inlet boundaries and normalized such that the value is 1 for the boundaries associated with that mixture and 0 for other boundaries. A mixture fraction is also associated with the evaporating spray droplets. With this convention, the sum of mixture fractions over all defined mixtures is unity. Since the mixture fractions sum to unity, K - 1 mixture fraction equations will have to be solved when K mixtures are defined.

Equation 4-19 is linear in fk and, therefore, also applies to linear combinations of the mixture fractions. The overall continuity equation is recovered by summing equation 4-19 over all mixtures. Let ik denote the mass fraction of the ith species in the kth mixture. It is easily shown that when equation 4-19 is multiplied by ik for each mixture fraction and summed over all mixture fractions, the following equation is obtained.

(4-20)

where:

(4-21)

This is the transport equation for the mass fraction of a non-reacting species, showing that composition can be calculated from the mixture fractions using equation 4-21 when the diffusion coefficients of all species are equal. The boundary conditions for the mixture fractions are defined

131


such that the boundary conditions for the mass fractions are satisfied by equation 4-21 as well. The effect of mass diffusivity differences among different species is negligible in most turbulent flows at moderate to high Reynolds numbers (convection-driven flows). The use of mixture fractions normally reduces the number of variables to be solved because the number of mixtures is usually less than the number of species.

Mixture fractions are also used with certain reaction models to calculate the composition of reacting flows.

Chemistry-Theory-Gas Phase Reaction Models

Chemistry Module Theory-Gas Phase Reaction Models-Introduction

The following gas phase reaction models are available:

Instantaneous Chemistry Model (for Mixture Mass Fraction approach)

Equilibrium Model (for Mixture Mass Fraction approach)

Finite-Rate Model (for Mixture Mass Fraction approach)

Finite-Rate Model (for Species Mass Fraction approach)

Chemistry Module Theory-Gas Phase Reaction Models-Instantaneous Chemistry Model

In the instantaneous chemistry model, the reactants (species on the left-hand side of equation 4-8) are assumed to react completely upon contact. The reaction rate is infinitely rapid and the reactants cannot exist at the same location. The following discussion will be limited to the case of two reactants, which are commonly referred to as fuel and oxidizer, and one reaction step. A surface (flame sheet) separates the two reactants. The rate of reaction is controlled by the rate at which reactants are transported to this surface.

The mass fractions of all species are only functions of the mixture fractions. The mass fractions for the instantaneous chemistry model are calculated by first using equation 4-21 to calculate the composition that would occur without the reaction. The unreacted composition, denoted by the superscript “u”, is given by

(4-22)

The change in composition due to the instantaneous reaction is then added to the unreacted mass fractions, as described below. This approach is valid when the mass diffusivities of all species are equal.

A stoichiometrically correct reaction step needs to be specified. Consider a single reaction between 1 (fuel) and 2 (oxidizer) to produce an arbitrary number of product species.

(4-23)

132

Chemistry Module

Since only one reaction is being considered, the subscript referring to the reaction step has been omitted. The reaction is a global step, so the stoichiometric coefficients do not have to be integers. For example,

C3H8 + 4.9 O2 = 2.9 CO2 + 0.1 CO + 3.9 H2O + 0.1 H2

The mass of species i produced per mass of fuel consumed by the reaction is:

(4-24)

The stoichiometric coefficients in equation 4-24 are for the overall reaction and, therefore, positive for product species and negative for fuel and oxidizer. Positive values of ri indicate production and negative values indicate consumption. The instantaneous reaction consumes either all the fuel or all the oxidizer, whichever is limiting. The amount of fuel that is consumed is:

(4-25)

The change in each species due to the reaction is proportional to the change in fuel, with the proportionality constant given by equation 4-24. The mass fraction of each species is then given by:

(4-26)

The right-hand side of equation 4-26 is only a function of the kth mixture fractions. k-1 transport equations must be solved for the mixture fractions. These equations have no source terms due to chemical reactions.

Chemistry Module Theory-Gas Phase Reaction Models-Equilibrium Model

In the equilibrium chemistry model, as in the instantaneous reaction model, the composition is determined from the solution of transport equations for mixture fraction variables. This model assumes chemical reactions are so fast that the mixture is in chemical equilibrium. The main difference between this model and the instantaneous model is that the user does not have to specify a stoichiometrically balanced reaction step. The composition is determined by minimizing the Gibbs energy of the system.

Chemical equilibrium is reached at constant temperature and pressure when the Gibbs energy is minimized. The Gibbs energy per unit mass of a system with N species is:

(4-27)

where:

133


i = the chemical potential of species i (or the particle molar Gibbs energy)

= the standard state chemical potential

p0 = is a reference pressure of 1 atmosphere.

Since the chemical potential is a function of temperature and pressure, the Gibbs energy is minimized at constant T and P for the right combination of ni. Elements must be conserved by the change in composition, which adds additional constraints to the system:

(4-28)

where:

aij = the number of atoms of element j in species i

M = the total number of elements in the system

bj = total number of moles of element j per unit mass

The composition that minimizes the Gibbs energy while satisfying the element balances is obtained by introducing the function:

(4-29)

The quantitiesj are termed Lagrangian multipliers. Since equation 4-28 must be satisfied to conserve elements, the second term on the right-hand side vanishes and the composition that minimizes also minimizes G. Differentiating equation 4-29 with respect to ni gives:

(4-30)

Differentiating equation 4-29 with respect to i gives:

(4-31)

Setting equations 4-30 and 4-31 equal to zero gives N + M equations to be solved to give the composition at chemical equilibrium. With some algebraic rearrangement, this yields the following:

134

Chemistry Module

(4-32)

(4-33)

Equation 4-32 is simplified as shown below:

(4-34)

Substituting equation 4-34 into equation 4-33 yield the following:

(4-35)

Thus we have M nonlinear algebraic equations for the unknown values of Zj. The values of bj are calculated from the mixture fractions using equation 4-21, giving:

(4-36)

An iterative Newton method is used to solve the system of equations for fixed values of pressure and temperature. The values of cj are calculated from equation 4-34. An updated temperature is calculated from static enthalpy and the new values of cj. New values of Rj, which depend on temperature, are calculated on each iteration. The iteration process continues until convergence is achieved.

Chemistry Module Theory-Gas Phase Reaction Models-Finite-Rate Model (for Mixture Solution)

135


In the finite-rate chemistry model, as the name implies, a single reaction proceeds at a finite rate. The reaction stoichiometry is specified in the same manner as in the instantaneous chemistry model (equation 4-23). The model is restricted to two reactant species. In addition to the stoichiometry, a rate expression must be specified. The primary difference between the finite-rate and instantaneous models is that the mass fraction of fuel is calculated by solution of a transport equation with a source term due to chemical reaction for the finite-rate model. The mass fractions of the other species are calculated from the mixture fractions and the mass fraction of fuel.

The molar production rate of species i due to the single-step reaction is:

(4-37)

An Arrhenius form (equation 4-12) is used for the reaction rate coefficient. The reaction is irreversible (i.e., the reverse rate coefficient is zero). As this is a global model, the concentration exponents do not have to be the same as the stoichiometric coefficients.

The transport equation for the mass fraction of fuel, Yi, is

(4-38)

Transport equations are solved for K - 1 mixture fractions and the mass fraction of fuel. The mass fractions of the other species are calculated by first calculating the composition of the unreacted mixture and then adding the change in composition due to the reaction.

(4-39)

where Y1 =( Y1)u - Y1 and ri is given by equation 4-24. The only difference between equation 4-

39 for the finite-rate chemistry model and equation 4-26 for the instantaneous chemistry model is that the mass fraction of fuel is calculated from a transport equation in the finite-rate model.

Transport equations for mass fractions of species other than fuel are not solved, but can be derived from the transport equations for the fuel mass fraction and the mixture fractions.

Chemistry Module Theory-Gas Phase Reaction Models-Finite-Rate Model (for Species Solution)

This finite-rate model allows for specification of single or multiple reaction steps (see equation 4-8) to model the process. This multi-step mechanism can be generally represented as:

(4-40)

136

Chemistry Module

The multi-step reaction model does not use the concept of mixture fractions that are used in the other chemistry models. Transport equations are solved for the mass fraction of Nsp species. The transport equation for species i is:

(4-41)

The diffusive flux of species i, Jij, includes ordinary diffusion driven by concentration gradients and, optionally, thermal diffusion driven by temperature gradients. The mass diffusivities of individual species do not have to be equal with this chemistry model. The production rate of species i,i , is given by equation 4-10.

The source term is linearized to improve convergence.

(4-42)

where the indices n and n+1 denote the iteration at which the corresponding quantity is evaluated. There are two methods available for the solution of equation 4-41. The first uses the full Jacobian array in equation 4-42 and couples the solution of all mass fractions in a point-iterative equation solver. The second method only uses the diagonal elements of the Jacobian array and solves each mass fraction equation sequentially with a whole field equation solver.

This chemistry model cannot be used with liquid spray because the mass source terms due to evaporation are not included in the transport equations.

Chemistry Module Theory-Surface Reaction Models

The surface reaction models allow the calculation of deposition, etching, or catalytic reaction at surfaces. The surface reaction provides a boundary condition for the mass fractions of species in the fluid, rather than a source term in the transport equations. The general form of the surface reaction considered in CFD-ACE+ is:

(4-43)

here:

aij = gas species stoichiometric coefficient

137


bij = adsorbed species stoichiometric coefficient

cij = bulk species stoichiometric coefficient

Ng = total number of gas-phase species

Ns = total number of adsorbed species

Nb = total number of bulk (deposited) species

For this reaction, the surface reaction rate may be expressed as:

(4-44)

where:

kfj = forward rates

krj = reverse rates

As seen from the above expression, the surface reaction rate is assumed to be independent of the concentration of the bulk species.

The gas-phase concentrations at the surface are expressed as:

(4-45)

and the surface concentrations are expressed as:

(4-46)

where:

= gas -phase mass density in kg/m3

= surface site density in kmol/m2

138

Chemistry Module

=

gas-phase mass fractions adjacent to the wall

= surface site fractions

The mass flux of reacting species to the surface (or away from the surface for species produced by the reaction) equals the rate at which the species is consumed (or produced) by the reaction on the surface.

A species flux balance at the reacting surface yields

(4-47)

(4-48)

where, the left-hand side of equation 4-47 is the diffusive flux of species i normal to the surface and the right-hand side of equation 4-47 is the production rate of species i per unit area of surface, on a mass basis. Equation 4-47 and equation 4-48 are solved by coupled Newton-Raphson iterations.

The reaction (mass) flux can be computed by using two different approaches, namely the sticking coefficient method and the general rate method. The sticking coefficient method evaluates the production rate based on sticking probability and precursor thermal flux, while the finite-rate chemistry uses the kinetic expression (see equation 4-44) to evaluate the reaction rate.

For sticking coefficient expression, surface reaction rate equation 4-44 becomes:

(4-49)

where sticking probability(The probability that a molecule will adsorb upon collision with the reacting surface. It is defined as the rate of adsorption divided by the collision frequency with the surface.) is expressed in Arrhenius from and the thermal flux of precursor species A is:

(4-50)

To fit into the format of equation 4-44, the above rate can be expressed as:

139


(4-51)

where:

(4-52)

For some surface reactions, the Arrhenius rate expression for the rate constant may need to be modified for surface coverage by some species. In such cases, the rate constant is modified in the following manner to account for surface coverage:

where Ki and Kf are the first and last surface species, ki, ki, and ki are the three coverage parameters, and Xk(n) is the surface site fraction of the kth surface species on site n.

Chemistry-Turbulence-Combustion Interaction

Chemistry Module Theory-Turbulence-Combustion Interaction-Introduction

The different turbulence models in CFD-ACE+ to model the Reynolds stresses and turbulent heat and mass fluxes with an eddy viscosity are described in the Turbulence Module. The effect of turbulence on chemical reaction and on composition dependent variables, such as density or temperature, must also be considered for turbulent reacting flows. It is not enough to average the transport equations for mass fractions in turbulent reacting flows in a manner analogous to the treatment of heat and mass transport in a non-reacting flow. Density and temperature are nonlinear functions of the mass fractions of each species. The average values of density and temperature cannot be calculated from the average value of the mass fractions. The joint probability density function (PDF) of composition is used to account for turbulence effects on reacting flow.

The joint composition PDF is a complete statistical description of the composition of the fluid at a single point in space and time. If the PDF is known, then the average value of any function of composition can be evaluated by multiplying that function by the PDF and integrating over the range of possible compositions.

(4-53)

140

Chemistry Module

where:

= the joint PDF of the N mass fractions at the position x and time t and

= an arbitrary function of the mass fractions

Favre-averaged quantities can be calculated by defining a Favre-averaged PDF:

(4-54)

The Favre-averaged form of the PDF is used in CFD-ACE+. The tilde will be omitted in the following discussion.

CFD-ACE+ uses an assumed PDF model for turbulent reacting flows. A parametric form of the PDF is assumed and the parameters in the model are related to variables governed by transport equations. The parametric form of the PDF used in CFD-ACE+ assumes the composition can be specified by a single mixture fraction and a single reaction progress variable. This assumption limits the reaction models available when the prescribed PDF models is used. A single-step instantaneous or finite-rate reaction can be used. The mass diffusivities of all species must be equal and no more than two mixtures can be defined.

The Turbulence Combustion Interaction section includes:

Determining PDF

Determining Averaged Variables

Operator Splitting

In Situ Adaptive Tabulation (ISAT)

Subgrid Linear Eddy Model

Application to Large Eddy Simulation

Turbulence-Combustion Interaction-Determining PDF

Chemistry Module Theory-Turbulence-Combustion Interaction-Determining PDF

In CFD-ACE+, the joint composition PDF is a function of a mixture fraction and a reaction progress variable. The reaction progress is defined as:

(4-55)

where Yf is the mass fraction of the fuel in the one step reaction and the minimum and maximum values are functions of the mixture fraction. The mixture fraction and reaction progress are assumed to be independent, so the two-dimensional PDF is a product of the two one-dimensional PDFs.

141


(4-56)

The one-dimensional PDFs have two parameters that are related to the average and variance of the mixture fraction or reaction progress. Transport equations are solved for the average and variance of the corresponding variable. (Note: A transport equation is solved for the average fuel fraction instead of the average reaction progress because the reaction progress is not well defined when no fuel or no oxidizer is present. The average reaction progress is calculated from the average fuel fraction and mixture fraction.)

The transport equations for the average mixture fraction and average fuel fraction are derived by averaging equation 4-19 and equation 4-38. The source term due to chemistry in equation 4-38 is averaged using the joint PDF. The transport equations for the variances of the mixture fraction and reaction progress include production terms caused by gradients in the average values, dissipation terms, and (for the reaction progress) a term due to chemical reaction.

(4-57)

(4-58)

In preceding equation, CD has the value of 2.

See Also

Reaction Progress PDF

Mixture Fraction PDF

Chemistry Module Theory-Turbulence-Combustion Interaction-Reaction Progress PDF

Two choices are available for the reaction progress PDF. One is the top-hat function described below for the mixture fraction PDF. The other is a tri-delta function with three possible values.

142

Chemistry Module

(4-59)

Chemistry Module Theory-Turbulence-Combustion Interaction-Mixture Fraction PDF

Two choices are available for the mixture fraction PDF: a top-hat and beta PDF. The top-hat PDF has uniform probability between a minimum and maximum mixture fraction, with discrete probabilities for mixture fraction values of 0 and 1.

(4-60)

The parameters for the top-hat PDF are given below, as functions of the average and variance of the mixture fraction.

143


144

Chemistry Module

The beta PDF is a continuous distribution defined between the values of 0 and 1.

(4-61)

The parameters for the beta PDF are:

145


(4-62)

Chemistry Module Turbulence-Combustion Interaction-Determining Averaged Variables

Variables such as species mass fractions, temperature, and density are functions only of the mixture fraction and reaction progress for the reaction models allowed with the prescribed PDF model. The average values of these variables are obtained by integrating the product of the instantaneous values of the variable of interest and the joint PDF of the mixture fraction and reaction progress over the range of mixture fraction and reaction progress.

(4-63)

(4-64)

Since the mixture fraction and reaction progress are independent variables and the PDF for the reaction progress only has discrete values the two-dimensional integrals can be evaluated as a sum of one-dimensional integrals. For example:

(4-65)

where c1, c2, and c3 are the probabilities of the reaction progress equaling 0, , and 1. The integrals are evaluated numerically for different values of the average mixture fraction before the transport equations are solved. During the solution of the transport equations governing the problem, the average values of variables are determined by linear interpolation from the stored data.

Chemistry Module Turbulence-Combustion Interaction-Operator Splitting

This capability allows chemical kinetics to be treated separately from convection and diffusion. The de-coupling of the chemistry from the convection and diffusion provides better convergence of the governing transport equations compared to traditional (sequential) finite volume flow solvers. This approach requires time steps that are smaller than the cell residence time, condition

146

Chemistry Module

which is easily satisfied when performing Large Eddy Simulations. The option of fast table look-up of integrated species increments (In Situ Adaptive Tabulation - ISAT) should be used to replace the expensive direct integrations required in the ODE solver. The tabulation algorithm already assumes Operator Splitting.

Chemistry Module Turbulence-Combustion Interaction-In Situ Adaptive Tabulation (ISAT)

For chemistry problems involving more than ten degrees of freedom, direct integration is an impractical solution to detailed kinetics simulations. One of the better alternatives relies on dynamic generation of look-up tables - In Situ Adaptive Tabulation (Pope, 1997). The tables are constructed during the actual reactive flow calculation and each entry represents a point from the composition space which is accessed in the calculation, forming an unstructured, adaptive discretization of the chemical manifold. The errors arising from the retrieval process are controlled with satisfactory success using the concept of regions of accuracy. The retrieval process comprises direct integration (in the early stages of the flow calculation) and search and extrapolation on the elements of the data structure constituted as a binary tree.

ISAT can be applied only if the operator-splitting approach is employed on the composition evolution equation, such that the effects of mixing, reaction and transport in physical space are treated in separate steps. The solution to the reaction equation from the initial condition:

is an unique trajectory in the composition space. Given a fixed time step t, the solution:

obtained by integrating the reaction equation is a mapping of the initial condition into the reacted value. Consequently, in the dynamically generated table, the reaction mapping values:

at particular tabulation points have to be stored. The location of the tabulation points in the composition space is dictated by the conditions in the flow field. In addition, information about the local properties of the chemical manifold is recorded, thus the change in the mapping values can be calculated from the displacements in the initial condition. The local properties of the manifold are reflected by the mapping gradient matrix:

defined as:

and by the higher order derivatives.

In the neighborhood of each tabulation point, different levels of approximation can be used. From storage and accuracy point of views, the zeroth order approximation is the cheapest and the least accurate. The optimal choice is the linear mapping approximation .The mapping gradient matrix is also related to the magnitudes of the local error in using the linear approximation in the tabulation point neighborhood. To the leading order, the local error can be estimated as = |BG|

where is the displacement from the originating point and B is a scaling matrix. I

147


In order to have a valid linear approximation, the error should be less than the specified tolerance tol which, in combination with the intrinsic properties of the mapping gradient matrix G, defines a region of accuracy for each tabulation point 0.

The region of accuracy is described by a hyper-ellipsoid having the length of the principal axes proportional with the tolerance error and inversely proportional with the singular values of the mapping gradient matrix (obtained from a singular value decomposition). The singular values tend to unity if the time step tends to zero. If the time step is very large, the compositions will be close to equilibrium and hence the singular values will tend to zero. To prevent unreasonably large principal axes, the smaller singular values are brought to 0.5. For each query point q around a tabulation record, an estimate of the hyper-ellipsoid of accuracy is obtained from the mapping gradient matrix constructed with the modified singular values. If the query point is outside the estimated ellipsoid of accuracy, but the error is still less than the prescribed tolerance, then the principal axes of the hyper-ellipsoid are modified such that the query point is included or is on the boundary of the ellipsoid. Although this procedure might introduce points that do not satisfy the error constraints, it does provide an adequate error control.

The table is built dynamically. For a given time step and a given tolerance, the chemistry module sends a query composition to the tabulation module, and the related mapping value is returned. The returned value is either extrapolated from a table record or is obtained by direct integration.

The data in the table is organized in a binary tree structure. The tree leaves each contain a record consisting of: a tabulation point, its reaction mapping vector and mapping gradient matrix (all fixed), the corresponding unitary matrix from the singular value decomposition of the mapping gradient matrix and the lengths of the principal axes of the current estimate of the hyper-ellipsoid of accuracy (last two entry modifiable to accommodate growth changes). The nodes of the binary tree contain the parameters of a cutting hyper-plane passing through the middle-point between the children (tabulation points) of the parent node and is perpendicular to the line described by the children. This information is used in the search process as detailed below.

For a given query composition q, the binary tree nodes are used to select the leaf that is likely to be the closest to q, by determining the position of query point with respect to each cutting plane. If q is within the estimated hyper-ellipsoid of accuracy, then using the linear approximation

the mapping value is returned. If the query is outside the estimated hyper-ellipsoid of accuracy, the mapping is determined by direct integration and local error is computed. If the error satisfies the tolerance constraint then the estimated hyper-ellipsoid of accuracy is grown to include the query point (see figure 4-1). Otherwise, the new query point is entered in the table as follows. The tree leaf with the tabulation point that was referenced in the query is replaced with a node with children 0 and q. The entries in the tree node are the parameters of the cutting plane between the two new children.

148

Chemistry Module

Illustration of the EOA Growth Process

Chemistry Module Turbulence-Combustion Interaction-Subgrid Linear Eddy Model

Accurate modeling of turbulent reacting flows demands the resolution of turbulence-chemistry interaction at all ranges of length and time scales. The linear eddy mixing subgrid model (LEM) explicitly distinguishes among the different physical processes of turbulent stirring, molecular diffusion, and chemical reaction at all scales of the flow through the introduction of a reduced one-dimensional description of the scalar field (Kerstein 1988).

Through this approach, it is possible to resolve all length scales of the scalar field, even for flows with relatively high Reynolds and Schmidt numbers with affordable computational cost. Along the one-dimensional array, detailed statistical representation of the scalar field, including both single and multi-point statistics, can be obtained. The key to the model performance lies in the manner in which the real physical mechanisms of turbulent mixing are represented. The molecular diffusion is treated explicitly by the solution of the diffusion equation along the linear domain,

(4-66)

where is the particular scalar under consideration and D is its diffusion coefficient. Thus, molecular diffusion is treated exactly, subject to the assumption that the statistics of a three-dimensional mixing process can be represented within the reduced dimensionality of the linear eddy model. In regions with chemical reactions, the chemical source term can also be treated explicitly by solution of,

(4-67)

where:

149


= the reaction rate.

Since the flow field is resolved in the one-dimensional domain, no modeling is required of the above processes described by equation 4-66 and equation 4-67 above.

The influence of turbulent stirring is modeled stoichastically and is carried out by random rearrangements of the scalar field along the domain. Each rearrangement event involves spatial redistribution of the scalar field within a specified segment of the linear domain. The size of the selected segment represents an eddy size, and the distribution of eddy sizes is obtained by applying the Kolmogorov scaling law. Physically, rearrangement of a segment of size l represents the action of an eddy size l on the scalar distribution. Thus, it is specified by two parameters: , which is a frequency parameter determining the rate of occurrence of the rearrangement events (stirring), and f(l), which is a pdf describing the size distribution (eddy size) of the segments of the flow which are rearranged. The values of these parameters are determined by recognizing that the rearrangement event induces a random walk of a marker particle on the linear domain. Equating the diffusivity of the random process with scaling for the turbulent diffusivity provides the necessary relationships to determine and f(l). For a high Reynolds number turbulent flow described by a Kolmogorov cascade, the result is (McMurtry et al. 1992):

(4-68)

(4-69)

where ReL is the Reynolds number based on the integral length scale, is the kinematic viscosity, is the Kolmogorov scale, and L is an integral scale.

The numerical algorithm for the scalar rearrangement or turbulent stirring process is carried out by the use of the triplet map. It involves the following steps: selecting a segment of the linear domain for rearrangement; making three compressed copies of the scalar field in that segment; replacing the original field by the three copies; and inverting the center copy.

An illustration of the triplet map is shown below, where the last figure shows the rearranged scalar field after acted on by molecular diffusion. The triplet map has several important features pertinent to the turbulent stirring process. First, the triplet map results in a tripling of the scalar gradients within a selected segment, analogous to the effects of compressive strain. Furthermore, a multiplicative increase in level crossings of a single scalar value results. This is analogous to the increase in surface area of a specified scalar value, a characteristic feature of turbulent mixing processes. In this manner, the most important features of turbulent mixing are accounted for with this mapping: the increase in surface area and the associated increase in the scalar gradient.

150

Chemistry Module

Triplet Map Illustration

With these parameters and mapping method specified, a stand-alone LEM model simulation is carried out as follows. The scalar field is first initialized along the linear domain in a manner consistent with the configuration under study. Along this domain, the effects of molecular diffusion and chemical reaction are implemented as a continuous process as described by equation 4-66 and equation 4-67. Then at randomly selected times governed by the rate parameter l, diffusion and reaction processes are interrupted by rearrangement events. The size of the domain to be rearranged is randomly selected from the pdf f(l) . This process continues until a specified time has elapsed.

Chemistry Module Turbulence-Combustion Interaction-Application to Large Eddy Simulation

The main element of the linear eddy sub-grid formulation is the implementation of a separate linear eddy calculation in each grid cell. This LEM model process is parameterized by the local Reynolds number based on grid size. Within each computational grid cell, the linear eddy simulation represents the turbulent stirring (described by equation 4-68 and equation 4-69), molecular diffusion (equation 4-66), and chemical reaction (equation 4-67) that occur at the small scales of the flow. Thus, differing to other sub-grid models which primarily use cell averaged random values to model the turbulence-chemistry interaction, the LEM sub-grid model directly resolves the turbulence-chemistry interaction down to the molecular diffusion scale of the flow (well below the grid size in most engineering applications) along the 1-D array of N. While fully resolved direct numerical simulations would require an array of dimension N3, the economy of using the linear eddy as a sub-grid model is apparent. Furthermore, the LEM model provides a detailed description of the small scale structure that is lacking in other parameterizations of mixing and reaction at unresolved scales.

151


Schematic illustration of LEM splicing events, where the 1-D elements represents the ongoing linear eddy calculation and the arrows indicate the components of convective flux

across the grid cell surfaces (McMurtry et al. 1993).

However, the implementation of LEM sub-grid model in LES requires another process to couple the sub-grid mixing process to the large-scale transport process responsible for convection across grid cell surface. This is achieved by splicing events, in which portions of the linear eddy domains are transferred to neighboring grid cells, as shown. The amount of material transferred across each cell boundary is determined based on the convective flux across the same cell surface, as computed from the resolvable grid scale velocity. These splicing events occur at a frequency with a time step comparable to the LES time step, which is much larger than the molecular diffusion time step governing the convection-diffusion-reaction process in each sub-grid.

Chemistry Module Limitations

The diffusion coefficient () is same for all the mixture fractions in the mixture fraction option of the Chemistry module.

The Instantaneous Chemistry Model is valid when the mass diffusivities of all species are equal.

The Single-Step Finite Rate Chemistry Model is applicable only to two reactant species.

The LEM sub-grid model is applicable only to the species option of the Chemistry module.

Chemistry-Implementation

152

Chemistry Module

Chemistry Module Implementation-Introduction

The Implementation section describes how to setup a model for simulation using the Chemistry Module. The Implementation section includes:


Model Setup and Solution - Describes the Chemistry Module related inputs to the CFD-ACE-Solver


Chemistry Module Implementation-Grid Generation

The Chemistry Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). Furthermore all grid cell types are supported (quad, tri, hex, tet, prism, poly).


Implementation-Model Setup and Solution

Chemistry Module Model Setup and Solution-Introduction

CFD-ACE+ provides the inputs required for the Chemistry Module. Model setup and solution requires data for the following panels:

Problem Type

Model Options

Volume Conditions

Boundary Conditions

Initial Conditions

Solver Control

Output

Chemistry Module Implementation-Model Setup and Solution-Problem Type


Select Chemistry to activate the Chemistry Module. The Chemistry Module is required for any simulation that involves the mixing or reacting of multiple gases. Whenever the Chemistry Module has been activated, you must also activate the Flow module. The Chemistry Module should not be activated with the Cavitation, Free Surface, or Two Fluid Modules.

Model Setup and Solution-Model Options

Chemistry Module Implementation-Model Setup and Solution-Model Options-Introduction

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. The Chemistry Model Options section includes:

153


Model Options-Shared

Model Options-Chem

Model Options-Chem-Gas Phase

Model Options-Chem-Liquid Phase

Model Options-Unsteady Combustion

Model Options-In Situ Adaptive Tabulation (ISAT)

Chemistry Module Implementation-Model Setup and Solution-Model Options-Shared

There are no settings under the Shared tab that affect the Chemistry Module. See Model Options for details.

Chemistry Module Implementation-Model Setup and Solution-Model Options-Chemistry

The model options for the Chemistry Module are located under the Chemistry tab. The Chemistry Media section's Media field contains a pull-down menu with two choices:

Gas Phase - Select the gas option to study gas related problems.

Liquid Phase - Select the liquid option to study electrochemistry problems.

The example below displays the Gas Phase section that appears when the Gas Phase option is selected.

Chemistry Module - Model Options Panel - Chemistry Tab

Chemistry Module Implementation-Model Setup and Solution-Model Options-Chem-Gas Phase

When you select Chem Tab->Chemistry Media->Gas Phase option, the Gas Phase section of the panel appears. It enables you select a pre-defined reaction mechanism to be applied to all of the fluid regions of the solution domain. You can specify reaction mechanisms in the Reaction Database Manager (see Database Managers).

154

Chemistry Module

Model Options - Chem Tab - Gas Phase

The Chem Tab-Gas Phase section's Solve For field contains a pull-down menu with two choices:

Mixture Mass Fractions

Species Mass Fractions (for gas phase reactions).

Mixture Mass Fraction

If you select Mixture Mass Fraction, you can apply any reaction mechanism that is Instantaneous, Equilibrium, or Finite-Rate (for Mixture Fraction Approach). Mixture Mass Fraction usually requires fewer transport equations than Species Mass Fraction. However, some models and fluid property options are not available for Mixture Mass Fraction, as shown in the following table. If you would like to use one of these models, you must activate Species Mass Fraction.

Species Mass Fraction

If you select Species Mass Fraction, you must select a Finite-Rate (for Species Fraction Approach) mechanism. Select Species Mass Fraction if you anticipate using one of the models available only for this approach during a later restart run.

Models for Mixture Mass Fraction and Species Mass Fraction Mass Transport

Mixture Mass Fraction Species Mass Fraction

Chemistry Chemistry

Finite-Rate (single step) Reactions

Finite-Rate (multi step) Reactions

Instantaneous Reactions Surface Reactions

Equilibrium Reactions Properties (viscosity, conductivity) by Kinetic Theory

155


Multi-Component Diffusion

Model Options-Chem-Gas Phase-Reaction Models

Chemistry Module Gas Phase Options - Nitrous NOX

The nitrous oxide mechanism for the production of NOX can be significant, even dominant, for lean flame conditions. The mechanism is initiated by the reaction . Production of NOX by the nitrous oxide pathway is modeled in CFD-ACE+ as a residence time dependent component and a prompt component (similar to thermal NOX production). The prompt component is the result of super-equilibrium of radicals in the flame region. The residence time dependent NOX production is modeled by:

(1)

where the reaction rate is determined from with A=2.0e7 and E/R=35000 in SI units. The constants were set by matching the detailed kinetics results from LSENS. The prompt component of nitrous NOX production is modeled by:

(2)

where the reaction rate is determined from with A=1.9e3 and E/R=16000 in SI units and the exponent a=0.45. The subscript b indicates concentrations that are determined from the amount of those species entering the cell before reaction occurs. The prompt component of nitrous NOX is turned on only if the prompt NOX model is also turned on.

Chemistry Module Gas Phase Options - Prompt NOX

Prompt NOX is formed in the flame region for hydrocarbon fuels primarily through reactions involving HCN. A global reaction for the production of NOX by the prompt mechanism derived by De Soete and further discussed by Pourkashanian, et al. is the basis for the model used in CFD-Ace+:

(1)

The reaction rate is determined from with A=5.0 and E/R=6000 in SI units. The concentration order ranges from 0 to 1 and is found as a function of the mole fraction from a curve fit of the graphical data given in Reference 3. is a correction factor that is a function of the local equivalence ratio, pressure, and the number of carbon atoms in the fuel. The concentrations of the fuel and are based on the amount of those species entering the cell before reaction occurs. The NOX production is proportional to the flame area in the cell rather than the cell volume.

Chemistry Module Gas Phase Options - Effects of Turbulent/Chemistry Interaction

156

Chemistry Module

The production of NOX, especially thermal NOX, increases exponentially with temperature. Because of the strong nonlinearity, significant inaccuracy may be introduced by using mean values of temperature and species concentrations in determining NOX source terms. The turbulent variations in these quantities can be accounted for by using a density-weighted probability density function (PDF) on the mixture fraction variable and/or the progress variable. A PDF on the mixture fraction is most important for diffusion flames and a PDF on the progress variable is most important for premixed flames. A combination of both is often best for partially premixed flames. The PDF formulation is limited to cases in which the mixture fraction of all the species can be determined from one conserved scalar (mixture fraction) and/or a progress variable. The progress variable ranges from 0 for unburnt mixtures to 1 for burnt mixtures. The PDF shape for the mixture fraction is assumed to be either a top-hat or Beta function. The PDF shape for the mixture fraction is assumed to be either a 3-Delta or a 5-Delta function. The Delta functions for the progress variable allow for efficient 2-D integration when both mixture fraction and progress variable PDFs are used.

The mean value of the NOX source term is evaluated at each cell using the prescribed PDF from

(1)

where is the NO source term, F is the mixture fraction, and P is density-weighted PDF of the mixture fraction. The mean density is found from

(2)

The assumed PDF shapes for the mixture fraction (top-hat or beta) are dependent on the mean mixture fraction (available from CFD-ACE+) and the variance of the mixture fraction. The variance is either read from the CFD-ACE+ Restart file, if available, or calculated by CFD-POST from the steady-state transport equation

(3)

where

(4)

The assumed PDF shapes (Delta functions) are dependent on the mean progress variable and the variance of the progress variable. The progress variable variance must be available from CFD-ACE+.

Chemistry Module Gas Phase Options - CO Post Processing

CO concentrations in 2-D or 3-D reacting flow fields are calculated by assuming that the deviation of the calculated CO field from the equilibrium value is small or that the calculated CO concentration is small so that the post-processed CO concentration has negligible effect on the heat release and the overall flow field. It is also assumed that equilibrium values of , CO and OH have been calculated by CFD-ACE+ for the Warnatz CO oxidation option or that equilibrium

157


values of , CO, , and have been calculated for the Dryer-Glassman CO oxidation option. The reaction in CFD-ACE+ may be either instantaneous or 1-step with equilibrium products. The CO field is solved by calculating the CO source term for each cell and using the convective and diffusive fields from CFD-ACE+ (from the .AFL file). The solution assumes that the upwind differencing scheme was used in CFD-ACE+ (See Mass Flow). CO is produced from the consumption of fuel. For example, consumption of 1 mole of (as predicted by CFD-ACE+) produces 3 moles of CO. The CO concentration is also constrained in the solution to be greater than or equal to the equilibrium CO concentration. The Warnatz option for CO oxidation reaction is given by

(1)

From this reaction, the destruction of CO can be expressed as

(2)

where the reaction rate is determined from with , , and

in SI units. The subscript e indicates equilibrium concentrations. The constant A has been modified in CFD-POST to a value of to better fit experimental results for practical combustors.

The oxidation of CO for the Dryer-Glassman option is given by

(3)

where the reaction rate is determined from with , , and

in SI units. The constant A has been modified in CFD-POST to a value of to better fit experimental results for practical combustors.

A similar model is also given by Howard et al. and additional work on CO oxidation is given by Baulch and Drysdale.

Chemistry Module Implementation-Model Setup and Solution-Model Options-Chem-Liquid Phase

When you choose Chem Tab->Chemistry Media->Liquid Phase, the Liquid Phase section of the panel appears.

158

Chemistry Module

Model Options - Chem Tab - Liquid Phase

Liquid Phase

The Liquid Phase section contains an Applications pull-down menu with the following options:

General Liquid Chemistry

o Solve Concentration - When you choose this option, you must then go to the Tools Menu->Database option. The Database Manager opens. Click the Species button and define your species. Click the Mixtures button. Under User Input, select the Concentration option. At the bottom of the window, select Enter Molar Concentration.

o Binary Diffusion

o Volume Reaction

General Liquid Chemistry Option

Biochemistry

o Binary Diffusion

o Volume Reaction

o Ionization

159


Biochemistry Option

Chemistry Module Implementation-Model Setup and Solution-Model Options-Unsteady Combustion

This model option is visible only for unsteady problems with Gas Phase media and the Species Mass Fraction approach selected. This feature is available only for CFD-ACE+ reaction sources. By activating Solve Combustion you can select a combustion model with the option to use ISAT in the calculation.

Laminar Chemistry with Operator Splitting is the default combustion model with the option Staggered Chemistry. This feature is very useful for accelerating the solution for large problems with complicated reaction mechanisms. By picking this option, the chemical rates are computed only once per time step at the last iteration and saved for the next time step.

When the Large Eddy Simulation closure Localized Dynamic is used you may choose another combustion model, the Sub-grid Linear Eddy Model, which is dependent on the sub-grid kinetic energy.

You may select the Integrated Mean Reaction LEM option for greater accuracy of the reaction rates calculation. This forces the chemistry module to integrate the reaction rates within each time sub-time step. Otherwise, the rates are calculated by simply calling the kinetic rate subroutines - a faster approach but less accurate.

To further speed up the simulation, you can set a Reaction Cutoff Temperature. In all cells of the computational domain for which the temperature is less than the cutoff value, the reactions rates are set to zero. The default value is 300K.

Chemistry Module Implementation-Model Setup and Solution-Model Options-In Situ Adaptive Tabulation (ISAT)

The ISAT algorithm may be chosen by checking Use ISAT for the available combustion models. ISAT reduces the number of direct integrations or full reaction rates calculations that are performed for each cell during the simulation. You can set several parameters for this algorithm.

The ISAT algorithm may optimized further by the use of multiple trees as function of temperature range and by controlling the size and their efficiency.

The temperature range can be divided in a number of Temperature Intervals such that each interval is represented by an ISAT tree. Specify the temperature range by setting the Maximum Temperature and Minimum Temperature values.

If you choose Scale Temperature, the Maximum Temperature value is also used for scaling the temperature variable, thus setting the error control level with respect to the [0,1] range.

The ISAT Tolerance parameter dictates the accuracy of the ISAT algorithm. The smaller the value the more direct integrations are performed.

160

Chemistry Module

The efficiency of the ISAT is controlled by setting the maximum number of records in the tree with Maximum Additions and the ISAT Threshold which represents the ratio of additions per number of queries. When these values are exceeded the tree is deleted and a new tree is built. This ensures that the root of the tree is situated closer to the center of the chemical manifold, resulting in a more balanced tree structure and hence greater efficiency.

In the case of SVD non-convergence, the maximum number of iterations for the Singular Value Decomposition algorithm can be increased with SVD Max. Iterations.

You may also choose between three kinds of ISAT algorithms differentiated by the type of extrapolation method used in the error control. The Full Algorithm uses linear extrapolation and growth of regions of accuracy. The Linear Extrapolation results in fixed regions of accuracy. The Zeroth Order Approximation uses direct values that were previously stored in the tree. The trade-off is again between speed and accuracy.

When ISAT is used with Sub-grid Linear Eddy Model and the Integrated Mean Reaction LEM option checked, the user may divide the sub-grid time step into several ISAT trees with LEM Time Intervals. The control over this value is not entirely in the possession of the user, in that the minimum number of time intervals is not known a priori and has to be set in an iterative manner.

Model Setup and Solution-Volume Conditions

Chemistry Module Implementation-Model Setup and Solution-Volume Conditions-Introduction

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. See Control Panel-Volume Conditions for details. Before any property values can be assigned, a volume condition entity must be made active by picking a valid entity from either the Viewer Window or the VC Explorer.

With the volume condition setting mode set to Properties, select any volume conditions and ensure that the volume condition type is set to Fluid. Only volume conditions that are of Fluid type need to have mixing properties specified (since there is no flow in solid or blocked regions there are no mixing properties for those regions.)

There are five volume condition properties required by the Chemistry Module; density, viscosity, specific heat, conductivity, and mass diffusion. The density and viscosity properties are discussed in detail in the Flow Module-Volume Conditions) and the specific heat and conductivity properties are discussed in detail in the Heat Transfer Module-Volume Conditions). The Volume Conditions section includes:

Volume Conditions-Mass Diffusion

Volume Conditions-Constant Schmidt Number

Volume Conditions-Constant Diffusivity

Volume Conditions-Mix Polynomial in T

Volume Conditions-Multi-Component Diffusion

Chemistry Module Implementation-Model Setup and Solution-Volume Conditions-Mass Diffusion

The options available for Mass Diffusion vary depending on whether the Mixture Mass Fraction or Species Mass Fraction approach has been selected (see Solution Method).

Chemistry Module Implementation-Model Setup and Solution-Volume Conditions-Constant Schmidt Number

161


Mass diffusion by a constant Schmidt Number can be used for both the Mixture and Species Mass Fraction approaches. When you specify a constant Schmidt Number (), the diffusion coefficient is calculated as:

(4-70)

Chemistry Module Implementation-Model Setup and Solution-Volume Conditions-Constant Diffusivity

You can specify a constant value of diffusion coefficient for a particular species using the Database Manager. Under the Database Manager, select the species of interest and under General tab, specify the value of diffusivity as coefficient c0.

Chemistry Module Implementation-Model Setup and Solution-Volume Conditions-Mix Polynomial in T

You can specify a fifth order polynomial for the variation of diffusivity as a function of Temperature. This is done in property manager under the General tab for each individual species. Coefficients c0, c1, c2, c3, c4 and c5 can be specified.

Chemistry Module Implementation-Model Setup and Solution-Volume Conditions-Multi-Component Diffusion

A more accurate model of the diffusive flux of each species is obtained by using the multi-component diffusion model. Multi-component diffusion can be activated only if you have selected mass transport by species mass fraction equations from the Model Options page (see Solution Method).

For the multi-component diffusion option, the species diffusive flux is split into two parts as shown below.

(4-71)

The first part is the concentration-driven diffusion and is calculated as:

(4-72)

The second part is the thermo-diffusion or Soret diffusion and is calculated as:

(4-73)

The concentration-driven diffusion coefficient is then calculated as:

162

Chemistry Module

(4-74)

where

Dij =

=

=

Lennard-Jones collision diameter

=

the collision integral

The collision integral, , is evaluated from the dimensionless temperature kB/ij

where

kB = Boltzmann constant

ij = characteristic energy of interaction,

Optionally, thermo-diffusion can be added to the concentration-driven diffusion by checking the Thermo-diffusion button. This option accounts for the species diffusion due to gradients of temperature. If this option is selected, the thermo-diffusion coefficient is calculated as:

(4-75)

where kij is the therm-odiffusion ratio.

163


The Multi-component Diffusion option also requires that you specify a method by which the program will satisfy species conservation i.e.:

(4-76)

There are three options:

None - The default of None means that the program will not strictly enforce species conservation for three or more species systems. For two species systems, the multi-component diffusion model does ensure species conservation

Reference Species - You can provide a Reference specie (usually the one with the large concentration) and the program then calculates the mass fraction of this reference species as 1.0 minus the sum of the remaining species concentrations.

Stefan-Maxwell - You can request the Stefan-Maxwell model in which the program uses an approximate form of the Stefan Maxwell equations to ensure species conservation.

Model Setup and Solution-Boundary Conditions

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Introduction

Click the Boundary Conditions [BC] tab to see the Boundary Conditions. See Control Panel-Boundary Conditions for details. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or the BC Explorer.

The Chemistry Module is fully supported by the Cyclic, Thin Wall, and Arbitrary Interface boundary conditions. (See Cyclic Boundary Conditions, Thin-Wall Boundary Conditions , or Arbitrary Interface Boundary Conditions for details).

The general boundary conditions for the Chemistry Module are located under the Chemistry tab and can be reached when the boundary condition setting mode is set to General. Each boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). The Chemistry Boundary Conditions section includes:

Boundary Conditions-Inlets

Boundary Conditions-Outlets

Boundary Conditions-Walls

Boundary Conditions-Surface Reactions

Boundary Conditions-Rotating Walls


Boundary Conditions-Interfaces

Boundary Conditions-Thin Walls


Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Inlets

164

Chemistry Module

For any inlet boundary condition, you must specify how to set the species concentration for each cell face on the boundary condition patch.

To set species concentration for an inlet boundary condition:

1. Select a previously defined mixture to be brought in at the inlet.

2. Under the Chemistry tab, select an option from the pull down menu (it lists all of the mixtures defined for the model).

3. If the mixture is not present, click the Define button to launch the Property Manager and define a new mixture.

See Database Manager-Mixtures for details on how to define a mixture.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Outlets

Under the Chemistry tab, specify a mixture for the outlet boundary condition just as you would for an inlet. This mixture will only be used where there is inflow through the outlet boundary condition.

Inflow through an outlet can occur anytime during the solution convergence process (even if the final solution indicates all outflow) so it is recommended that you supply a reasonable mixture definition. If the final solution shows inflow through an outlet boundary condition, this indicates that the boundary condition may not have been located in an appropriate place. When this happens, an unphysical solution and convergence problems may result and you should relocate the outlet boundary condition to an area where there is total outflow if possible.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Walls

Various boundary conditions can be specified under the Chemistry tab. They are:

Zero Flux - Flux of all species to the walls is set to zero

Fixed Mixture - You can specify the mixture composition on all the cells adjacent to the wall. You must ensure that SUMMATION = 0. This mixture can be defined in the property manager and will show up in the list of mixtures available for this boundary. This option is valid only when Liquid is selected.

Species Specification - This is similar to the Fixed Mixture boundary condition. You can pick each one of the species available in a mixture and choose one of the four evaluation techniques in Evaluation method for that particular species. This option is valid only when Liquid is selected.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Surface Reactions

You can specify a surface reaction mechanism to be applied to each wall boundary condition. Under the Surface Reactions tab is a pull-down menu that lists all of the surface reaction mechanisms defined for the model. If the mechanism is not present, press the Define button to launch the Surface Reaction Manager and define a new mechanism. See Database Manager- Surface Reaction Mechanisms for details on how to define a surface reaction mechanism.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Rotating Walls

165


The Chemistry Module boundary condition specifications for rotating walls are identical to that as described for wall boundary conditions. They are:

Zero Flux - Flux of all species to the walls is set to zero

Fixed Mixture - You can specify the mixture composition on all the cells adjacent to the wall. You must ensure that SUMMATION = 0. This mixture can be defined in the property manager and will show up in the list of mixtures available for this boundary. This option is valid only when Liquid is selected.

Species Specification - This is similar to the Fixed Mixture boundary condition. You can pick each one of the species available in a mixture and choose one of the four evaluation techniques in Evaluation method for that particular species. This option is valid only when Liquid is selected.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Symmetry

The symmetry boundary condition is a zero-gradient condition. Species are not allowed to cross the symmetry boundary condition. There are no Chemistry Module related values for symmetry boundary conditions.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Interfaces

The interface boundary condition allows two computational regions to communicate information. If the interface boundary condition lies between a fluid volume condition and a solid volume condition, then you may specify a surface reaction mechanism to be applied to that location.

Interface boundary conditions can be converted to Thin Walls. See Thin-Wall Boundary Conditions and Arbitrary Interface Boundary Conditions for information on other ways for computational domains to communicate.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Thin Walls

The Thin Wall boundary condition is fully supported by the Chemistry Module. See Thin-Wall Boundary Conditions for instructions on how to setup a thin wall boundary condition.

The Chemistry Module treats a thin wall boundary condition the same as a wall boundary condition (see Walls). Under the Chemistry tab, there are inputs available for surface reaction specification if surface reactions have been activated.

Chemistry Module Implementation-Model Setup and Solution-Boundary Conditions-Cyclic

The Cyclic boundary condition is fully supported by the Chemistry Module. See Cyclic Boundary Conditions to learn how to setup a cyclic boundary condition. There are no Chemistry Module related settings for the cyclic boundary condition.

Chemistry Module Implementation-Model Setup and Solution-Initial Conditions


You can specify the Initial Conditions as constant values or read from a previously run solution file. If you specify constant values, you must provide an initial mixture as required by the

166

Chemistry Module

Chemistry Module. The mixture definition can be found under the Chemistry tab. The mixture definition must be previously defined using the Property Database Manager. See Database Manager-Mixtures for details on defining mixtures.

Although the Initial Condition mixture does not affect the final solution, a reasonable mixture should be specified so that the solution does not have convergence problems at start-up.


Chemistry Module Implementation-Model Setup and Solution-Solver Control-Introduction

Click the Solver Control [SC] tab to see the Solver Control Panel and obtain access to the settings that control the numerical aspects of the CFD-ACE-Solver and output. It includes:

Spatial Differencing Scheme

Solver Selection

Under-Relaxation Parameters

Variable Limits

Chemistry Module Implementation-Model Setup and Solution-Solver Control-Spatial Differencing Scheme

Under the Spatial Differencing tab, select the differencing method to be used for the convective terms in the equations. Activating the Chemistry Module enables you to set species or mixture mass fraction calculations. The default method is first order upwind. See Control Panel-Solver Controls-Spatial Differencing Scheme for more information on the different differencing schemes available. See Numerical Methods for numerical details of the differencing schemes.

Chemistry Module Implementation-Model Setup and Solution-Solver Control-Solver Selection

Under the Solvers tab, select the linear equation solver to be used for each set of equations. Activating the Chemistry Module enables you to set the mixture or species mass fraction equations. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 50 sweeps. See Solver Selection for more information on the different linear equation solvers available and Linear Equation Solvers for numerical details of the linear equation solvers.

Chemistry Module Implementation-Model Setup and Solution-Solver Control-Under Relaxation Parameters

Under the Relaxation tab, select the amount of under-relaxation to be applied for each of the dependent (solved) and auxiliary variables used for the equations. Activating the Chemistry Module enables you to set the mixture or species mass fraction dependent variables. See Under Relaxation Parameters for more information on the mechanics of setting the under relaxation values. See Under Relaxation for numerical details of how under-relaxation is applied.

The mixture or species mass fraction equations use an inertial under relaxation scheme and the default values are 0.2. Increasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.


167


Chemistry Module Implementation-Model Setup and Solution-Solver Control-Variable Limits

Under the Limits tab, select the settings for minimum and maximum allowed variable values. CFD-ACE+ will ensure that the value of any given variable will always remain within these limits by clamping the value. Activating the Chemistry Module enables you to set limits for the mixture or species mass fraction variables. See Control Panel-Solver Controls-Variable Limits for more information on how limits are applied.

The default minimum and maximum limits for the mixture or species mass fractions are 0 and 1 respectively. These limits should not be changed or an unphysical solution may result.

Chemistry Module Implementation-Model Setup and Solution-Solver Control-Advanced Settings

Advanced Settings

Shared

Buffered Output

Higher Accuracy

Chem

There are three settings under the advanced options tab: Cut Diffusion (Chem) at Inlets, CFL Relaxation, and Species Conservation Enforced.

The Inlet Diffusion option allows you to disable the species diffusive link to an inlet boundary. For low pressure transport problems this may be important because it allows you to prevent the diffusive loss of species through an inlet and gives you better control over the amount of each species in the domain since you only have to account for inlet convection.

When using CFL based relaxation, an effective time step is calculated for each computational cell (local time stepping). The size of the cell’s effective time step is calculated by determining the minimum time scale required for convection, diffusion, or chemistry to occur in that cell. This minimum time scale is then multiplied by a user input factor to determine the final effective time step which will be used for that cell.

The default inertial relaxation method can be switched to the CFL based relaxation method by going to SC-->Adv and checking the appropriate check boxes for each module. The relaxation factor defined in SC-->Relax is used as the CFL multiplier.

Rule of Thumb: Inverse value of the usual inertial relaxation factor.

Effect of Value:

5 = Default Value




The Species Concentration Enforced option is intended for use with PEM fuel cell cases, but could be used for other applications. For multicomponent diffusion problems, it is recommended that the Stefan-Maxwell enforcement method be used for species conservation.


168

Chemistry Module

Chemistry Module Implementation-Model Setup and Solution-Output-Output Introduction

There are no settings under the Output tab that effect the Chemistry Module. See Control Panel-Output Options for details about the available output settings.

The Output section includes:

Graphical Output

Printed Output

Chemistry Module Implementation-Model Setup and Solution-Output-Graphical Output

Under the Graphics tab, select the variables to output to the graphics file (modelname.DTF). These variables will then be available for visualization and analysis in CFD-VIEW. Activating the Chemistry Module provides output of the variables listed in the table below:

Chemistry Module Graphical Output

Variable Units

Mixture or Species Mass Fractions -

Reaction Rate (if gas phase reactions are present)

kg/m3-s

Deposition Rate (if surface reactions are present)

kg/m2-s

Species Flux kg/m2-s

Species Diffusivity m2/s

Species Thermodiffusivity m2/s

Chemistry Module Implementation-Model Setup and Solution-Output-Printed Output

Under the Print tab, select the printed information to be written to the text based output file (modelname.out). Activating the Chemistry Module enables you to set the output of a species summary. See Control Panel-Printed Output for more information on the general printed output options including boundary condition integral output and monitor point output.

The species summary will provide a tabulated list of the integrated mass flow (kg/s) through each flow boundary (inlets, outlets, interfaces, etc.) for each species.

169


In addition to the printed species summary, you can select gas phase species flux information at reacting surfaces for one-way coupling to feature scale models. This coupling is only available when you use the Species Mass Fraction solution approach, and is activated by choosing the Feature Scale Coupling option under the Print tab. The locations and format of the output may be specified either through the User Input option, or by requesting that a text file be read. Next, specify the locations of the link points and the format for the flux data. The resulting data is printed to files named modelname.nnn.FSC, where nnn is the link point number. The available formats are:

Generic: the actual and maximum possible fluxes of each species to the surface, in units of #/cm2 sec, are provided. The maximum flux is the appropriate input for a feature scale model, the actual flux is provided to allow estimation of effective sticking coefficients. The coordinates of the actual computational face center at which the fluxes were obtained, and the temperature of that face, are also provided.

EVOLVE: the operating condition lines of the EVOLVE input deck are provided for the gas phase species participating in a surface reaction at the link points. The first line of the output is the temperature (Kelvin) and the pressure (Torr). The next line consists of the EVOLVE 'ioper' flags for each species, with each value set to 3 to signify that the operating condition input is fluxes in gmole/cm2 sec. The final line is the maximum fluxes of the species to the surface . An additional file, modelname.EVSPEC, is written to provide the species output order.

SPEEDIE: the general SPEEDIE output format is equivalent to the Generic format, providing the user the flexibility of choosing which species correspond to the SPEEDIE DEPO, ION, or CHEM species.

SPEEDIE LPCVD1 and LPCVD2: the user specifies the species in the ACE model that will correspond to the DEPO (and DEPO2) species in the SPEEDIE LPCVD 1 (2) model, and additionally species the substrate and deposited materials. The modelname.nn.FSC file contains the fluxes, deposited material, and model parameter data for simulation of low pressure chemical vapor deposition using the corresponding SPEEDIE model.

For simulations using the plasma module, additional data is provided if a sheath model is specified at the reacting surface. In this case, the energy and angular distribution functions are written for each ion at the surface in files named modelname.(e,a)df.species_name.nnn.dat. These files provide probability distributions for the energy and angular distributions of the ions striking the surface, with the energies in electron volts and the angles in degrees. If you select EVOLVE format output, angular flux distribution files in EVOLVE format named modelname.species_name.nn.EVFLX are also provided. Similarly, if you select the SPEEDIE format, the SPEEDIE format *.mo files with energy and angular distributions for the fluxes of each species are provided.

Chemistry Module Implementation-Post Processing

CFD-VIEW can post-process the solutions. When the Chemistry Module is invoked, the mixture or species mass fraction fields are usually of interest. You can view these fields with surface contours and analyze them using point and line probes.

For reacting problems (gas phase or surface chemistry) output of the reaction rate and/or deposition rate are usually of interest. The deposition rate is only written on the surfaces for which a surface reaction has been applied, and is therefore best analyzed through the use of point or line probes. A complete list of post processing variables available as a result of using the Chemistry Module are shown in the table.


170

Chemistry Module


Nox_Rate NOx production rate kg/(m3-sec)

Progress Progress Variable -

React_Rate Reaction Rate kg/(m3-sec)

Species name Species Mass fraction -

The species summary written to the output file (modelname.out) is used to determine quantitative results. The species summary can also be used to judge the convergence of the simulation. Due to the law of conservation of mass, the summation of all species flowing into and out of the computational domain should be zero (unless species sources or sinks such as gas phase and surface reactions are present). In the simulation a summation of exactly zero is almost impossible, but you should see a summation that is several orders of magnitude below the total species inflow.

Chemistry Module Frequently Asked Questions

What information does the CVD file contain? What is the file format?

The CVD file is written when the Chemistry module is activated and surface reactions are occurring. The deposition/etch rate can be obtained for the reacting surface. The deposition/etch rate is provided at every boundary face on the reacting surface. In the example CVD file below, xf is the x location of the face center, yf is the y location of the face center, Dep/Etch Rate is the deposition/etch rate at each boundary face. and SumYw-1 is the summation of the mass fractions at the wall minus one. Note that the deposition/etch rate is reported in microns/min.

Dep(-ve)/Etch(+ve) rate in microns/min

xf yf Dep/Etch Rate SumYw-1

-0.300000E-01 0.100000E-01 0.000000E+00 0.2400E-07

-0.300000E-01 0.235584E-01 0.000000E+00 0.2394E-07

-0.300000E-01 0.385712E-01 0.000000E+00 0.2380E-07

I have specified bulk species as a product in my reaction, but I do not see and deposition/etch rate?

When using bulk species in a reaction, the bulk species must be the first product species in the reaction. If the bulk species is not the first product species in the reaction, then you will not see any deposition or etching (given as a deposition or etching rate in CFD-VIEW). Here is an example of a correct reaction and an incorrect reaction:

A + B C(B) + D (Correct)

A + B D + C (B) (Incorrect)

171


172

Chemistry Module Examples

The following tutorial uses the Flow, Heat Transfer, and Chemistry Module exclusively:

Surface Reaction in a 2-D Reactor

The following tutorials use the Flow, Heat Transfer, and Chemistry Module in conjunction with one or more other modules:



Chemistry Module References

Kerstein, A. R., "A Linear Eddy Model of Turbulent Scalar Transport and Mixing." Combust. Sci. Tech. 60, p. 391, 1988.

McMurtry, P. A., Menon, S., and Kerstein, A. R., "A Linear Eddy Sub-Grid Model for Turbulent Reacting Flows: Application to Hydrogen-Air Combustion." Twenty-Four Symposium (international) on Combustion, The Combustion Institute, pp. 271-278 (1992).

Pope, S.B., (1997), “Computationally Efficient Implementation of Combustion Chemistry Using In Situ Adaptive Tabulation.” Combustion Theory and Modeling, Vol. 1, pp. 41-63.

Pratt David T., Wormeck, John J., CREK, A computer program for calculation of Combustion Reaction Equilibrium and Kinetics in Laminar or Turbulent Flow. Thermal Energy Laboratory Department of Mechanical Engineering, Washington State University Report WSU-ME-TEL-76-1.Pullman, WA: 1976.

Somorjai, G. A., Introduction to Surface Chemistry and Catalysis, Wiley-Interscience, New York, 1994

User Scalar Module

User Scalar Module Introduction

The User Scalar Module enables you to compute the transport of scalars. Activating the User Scalar Module implies the solution of one or more scalar variables (by solving a general transport equation for each requested scalar). This capability is often used with one or more of the other CFD-ACE+ modules to provide a multi-physics based solution to an engineering problem (such as coupling a user scalar with flow, heat transfer, mixing, etc.). The user scalar can be passive (i.e., it does not affect any other solution variable), or active (i.e., other solution variables are affected by the scalar field). The User Scalar Module includes:

User Scalar-Applications

User Scalar-Features

User Scalar-Theory

User Scalar-Limitations

User Scalar-Implementation

User Scalar-Frequently Asked Questions

User Scalar-References

User Scalar Module Applications

The User Scalar Module enables you to model any passive or active scalar quantity. The more common applications are for electric potential, electromagnetic fields, and inert/passive chemical species (tracers).

User Scalar Module Features

The User Scalar Module has many features which may or may not be activated for a simulation.

Scalar Types

The User Scalar Module provides a solution to almost any type of scalar problem. There are three classes of scalar variables: generalized scalar, passive scalar, and Poisson scalar. The difference between each is the transport mechanism that is allowed in fluid and solid regions. The differences are summarized in the table.

General Scalar - For a general scalar, in addition to convective and diffusive transport in the fluid phase, the diffusional transport of the specified scalar inside solids is included in the computation.

Passive Scalar - Passive scalars do not affect the velocity, thermal, or any other computed field. Transport of such scalars inside solids (convective or diffusive) is not permitted.

Poisson Scalar - For Poisson scalars, diffusion is the only mechanism of transport in both solids and fluids. Convective effects are turned off in computing their transport. The

173


density is taken out of the generic transport equation. One typical example of this type of scalar is the electric potential.

Transport Mechanisms for Different Types of User Scalars

Transport Mechanism

in Fluid Volumes Transport Mechanism

in Solid Volumes

Scalar Type

Convection Diffusion Convection Diffusion

General yes yes N/A yes

Passive yes yes N/A no

Poisson no yes N/A yes

Scalar Control

You can control the behavior of user scalars through the user subroutines. User subroutines enable you to modify the user scalar source terms, boundary conditions, and diffusivity. This allows you to couple the user scalar equations with other equations in your simulation. See User Subroutines for more information.

User Scalar Module Theory

The generic transport equation of a user scalar is written as:

(5-1)

where D is the diffusivity and S is the volumetric source term.

For all the scalars, the boundary conditions are generalized as:

(5-2)

where n denotes the normal direction at the boundary. You can choose appropriate values for the three coefficients A, B, and C to specify the desired boundary conditions (see Boundary Conditions).

174

User Scalar Module

At a solid/solid or fluid/solid interface, the diffusive flux normal to the boundary is conserved:

(5-3)

User Scalar Module Limitations

The User Scalar module is currently not compatible with the Free Surfaces Module (VOF).

User Scalar-Implementation

User Scalar Module Implementation-Introduction

The Implementation describes how to setup a model for simulation using the User Scalar Module. It includes:

Grid Generation - Describes the types of grids that are allowed and general gridding guidelines.

Model Setup and Solution - Describes the User Scalar Module related inputs.

Post Processing - Provides tips on what to look for in the solution output.

User Scalar Module Implementation-Grid Generation

You can apply the User Scalar Module to any geometric system (3D, 2D planar, or 2D axisymmetric). All grid cell types are supported (quad, tri, hex, tet, prism, poly).



User Scalar Module Implementation-Model Setup and Solution-Introduction

CFD-ACE+ provides the inputs required for the User Scalar Module. Model Setup and Solution requires data for the following panels:

Problem Type

Model Options

Volume Conditions

Boundary Conditions

Initial Conditions

Solver Control

Output

175


User Scalar Module Implementation-Model Setup and Solution-Problem Type


Select User Scalar to activate the User Scalar Module. The User Scalar Module can work with any of the other modules in CFD-ACE+.

User Scalar Module Implementation-Model Setup and Solution-Model Options

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. All of the model options for the User Scalar Module are located under the User Scalar (Scalar) tab.

Shared Tab

There are no settings under the Shared tab that directly affect the User Scalar Module.

Scalar Tab

This panel enables you to specify the number of user scalars and the type and name of each user scalar. The steps for defining this information are given below.

Model Options - Scalar Tab

1. Enter the number of scalars in the Total Scalars field and click OK.

2. In the Current Scalar field, enter the scalar number that you want to make current, or use the arrow key at the far end of the field to specify which scalar is current.

3. For the current scalar, assign the type (see Scalar Types for detailed descriptions of each user scalar type).

To create a General Scalar, activate both Convection and Diffusion in Solid.

To create a Passive Scalar, activate only Convection.

To create a Poisson Scalar, activate only Diffusion in Solid.

4. Enter a name for the current scalar in the Scalar Name field.

5. Proceed with step 2 for every scalar.

User Scalar Module Implementation-Model Setup and Solution-Volume Conditions

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. See Control Panel-Volume Conditions for details. Before any property values can be assigned, one or more volume condition entities must be made active by picking valid entities from either the Viewer Window or the VC Explorer.

176

User Scalar Module

You can specify general scalar sources by changing the volume condition setting mode to Scalar. (see Source Term Linearization for details on setting general sources).

With the volume condition setting mode set to Properties select any volume conditions. There are three volume condition properties required by the User Scalar Module; density, viscosity and scalar diffusivity. Density is only used by the User Scalar Module only for user scalars which are of the General and Passive type. The viscosity property is used by the User Scalar Module only if the scalar diffusivity is to be calculated by the Schmidt number approach. The density and viscosity properties are discussed in detail in the Flow Module (see Volume Conditions).

A typical input panel for scalar diffusivity is shown.

Volume Condition Inputs for Scalar Diffusivity

The Total Scalars field lets you know how many user scalars have been defined (see Model Options). You must set each user scalar’s diffusivity as follows.


2. For the current scalar, pick the evaluation method to be used to calculate the scalar diffusivity (D). There are three choices available:

Constant-D = value specified in m2/s.

Schmidt Number - D = /Sc

User Sub (udiff_scalar) - D is defined by a user subroutine (udiff_scalar). Please see User Subroutines for details.

3. Enter the value of diffusivity or Schmidt number as appropriate.


User Scalar Module Implementation-Model Setup and Solution-Boundary Conditions


The User Scalar Module is fully supported by the Cyclic, Thin Wall, and Arbitrary Interface boundary conditions. (See Cyclic Boundary Conditions, Thin-Wall Boundary Conditions or Arbitrary Interface Boundary Conditions for details on these types of boundary conditions and instructions for how to implement them.)

All of the general boundary conditions for the User Scalar Module are located under the Scalar tab and can be reached when the boundary condition setting mode is set to General. Each

177


boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). See BC Type for details on setting boundary condition types.

The User Scalar Module differs from the other modules in the fact that the boundary condition for a user scalar has been generalized. The method described below works for the following boundary condition types:

Inlets

Outlets

Walls

Rotating Walls

Boundary conditions which are of type Symmetry will always have a zero gradient condition applied for the user scalar equations. Boundary conditions which are of type Interface will have a matching flux condition (see equation 5-3).

The generalized boundary condition for user scalars is evaluated according to the following equation:

(5-4)

You are required to assign values to the coefficients a, b, and c. Different boundary effects can be accomplished by the choice of coefficients a, b, and c and are summarized below.

Boundary Condition Coefficient Settings for Different Effects

Coefficient Setting Desired Effect

a b c

Result

Fixed Value (Dirichlet)

a = 0

c = c

Fixed Flux (Neumann)

b = 0 c = c

Flux as Function of Value (Combined)

c = c

A typical input panel for any user scalar boundary condition is shown.

178

User Scalar Module

Boundary Condition Inputs for User Scalar Coefficients

The Total Scalars field lets you know how many user scalars have been defined (see Model Options). You must set each user scalar’s boundary condition coefficients:


2. For the current scalar, pick the evaluation method to be used to specify the c coefficient. The choices are Constant value or User Defined c (see User Subroutines for details).

3. Enter the values for coefficients a, b, and c keeping in mind equation 5-4 and the information in table 5-2.


User Scalar Module Implementation-Model Setup and Solution-Initial Conditions

Click the Initial Conditions [IC] tab to see the Initial Conditions Panel. See Control Panel-Initial Conditions for details.

The Initial Conditions can be specified as constant values or read from a previously run solution file. If constant values are specified then you must provide initial values required by the User Scalar Module. The values are under the Scalar tab and a value must be set for every user scalar variable.


User Scalar Module Implementation-Model Setup and Solution-Solver Control-Introduction

Click the Solver Control [SC] tab to see the Solver Control Panel. See Control Panel-Solver Control for details.

The Solver Control panel provides access to the settings that control the numerical aspects of the CFD-ACE-Solver and output options. The User Scalar Module is different than the other modules in that all of the numerical controls are located on the Solver Control page under the Scalar tab.

179


Numerical Control Settings for User Scalar Variables

The mechanics of setting the numerical control parameters is the same for all four parameters (Solver, Spatial Differencing, Under Relaxation, and Limits). Each parameter is displayed in its own region, and the instructions below should be followed for every numerical control parameter:

The Total Scalars field lets you know how many user scalars have been defined (see Model Options). You must set each user scalar’s numerical control parameters as follows.

1. In the Current Scalar field, type in the scalar number that you want to make current, or use the arrow key at the far end of the field to specify which scalar is current.

2. For the current scalar, assign an evaluation method and/or values as appropriate.


The Solver Control section includes:



180

User Scalar Module


Solver Control-Variable Limits

User Scalar Module Implementation-Model Setup and Solution-Solver Control-Solver Selection

In the Solvers tab, you may select the linear equation solver to be used for each user scalar equation. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 50 sweeps and a convergence criteria is 0.0001. See Solver Selection for more information on the different linear equation solvers available and Linear Equation Solvers for numerical details of the linear equation solvers.

User Scalar Module Implementation-Model Setup and Solution-Solver Control-Spatial Differencing Scheme

Under the Spatial Differencing tab you may select the differencing method to be used for the convective terms in the user scalar equations. The default method is first order Upwind. See Control Panel-Spatial Differencing Scheme for more information on the different differencing schemes available. Also see Numerical Methods-Discretization for numerical details of the differencing schemes.

User Scalar Module Implementation-Model Setup and Solution-Solver Control-Under Relaxation Parameters

In the Under Relaxation region you may select the amount of under-relaxation to be applied for each of the solved user scalar variables. See Numerical Methods-Under Relaxation for numerical details of how under-relaxation is applied.

The user scalar equations use an inertial under relaxation scheme and the default values are 0.2. Increasing this value applies more under-relaxation and therefore adds stability to the solution at the cost of slower convergence.

The default values for all of the under relaxation settings will often be sufficient. In some cases, these settings will have to be changed, usually by increasing the amount of under relaxation that is applied although if the solution of the scalar equation is relatively simple, smaller values may be used to increase the convergence rate. There are no general rules for these settings and only experience can be a guide.

User Scalar Module Implementation-Model Setup and Solution-Solver Control-Variable Limits

Settings for minimum and maximum allowed variable values are in the Limits region. CFD-ACE+ will ensure that the value of any variable will always remain within these limits by clamping the value.

User Scalar Module Implementation-Model Setup and Solution-Output

There are no settings under the Output tab that affect the User Scalar Module. See Control Panel-Output Options for details. All scalars and the associated diffusivity coefficients are output by default.

Printed Output

181


Under the Print tab on the Solution Control page, select the printed information to be written to the text based output file (modelname.out). Activating the User Scalar Module allows output of a scalar flux summary in addition to the general printed output options. See Control Panel-Printed Output for details on the general printed output options including boundary condition integral output and monitor point output).

The scalar flux summary will provide a tabulated list of the integrated scalar flux (scalar unit-kg/s) through each flow boundary (inlets, outlets, interfaces, etc.).

Graphical Output

Under the Graphics tab, you can select the variables to output to the graphics file (modelname.DTF). These variables will then be available for visualization and analysis in CFD-VIEW. Activating the User Scalar Module allows output of the variables listed:

Scalar Module Graphical Output

Variable Units

Density kg/m3

Scalar Values scalar units

Scalar Diffusivity

kg/m-s (for passive or general scalar) m2/s (for poisson scalar)

User Scalar Module Implementation-Post Processing

CFD-VIEW can post-process the solutions. When you activate the User Scalar Module, the scalar fields can be seen with surface contours and analyzed through the use of point and line probes. A complete list of post processing variables available as a result of using the Scalar Module is shown in the table.



D_ScalarName Scalar Diffusion Coefficient kg/m-s

ScalarName Scalar Name -

The scalar flux summary written to the output file (modelname.out) is often used to determine quantitative results. The scalar flux summary can also be used to judge the convergence of the simulation. Due to the law of conservation of flux, the summation of all scalar flux into and out of the computational domain should be zero (unless scalar sources or sinks are present). In the simulation a summation of exactly zero is almost impossible, but you should see a summation that is several orders of magnitude below the total scalar flux inflow.

182

User Scalar Module

183

User Scalar Module Frequently Asked Questions

How do I fix the value of my user scalar at a boundary?

Set the generalized boundary condition coefficients to:

a = 0, b = 1, c = desired value.

How do I fix the flux of my user scalar at a boundary?

Set the generalized boundary condition coefficients to:

a = 1, b = 0, c = desired flux.

User Scalar Module References

Versteeg, H.K. and Malasekera, W., 1995, An Introduction to Computational Fluid Dynamics." John Wiley & Sons Inc, New York, pp24.

Radiation Module

Radiation Module Introduction

The Radiation Module enables you to solve radiation problems. Electromagnetic radiation is emitted by all substances due to the changes in the internal molecular and atomic energy states. The wavelength of electromagnetic radiation ranges from very long radio waves to very short cosmic rays. The visible light is in a narrow range from 0.4 - 0.7 m and thermal radiation is in the infrared range. One important difference between radiation and other modes of heat transfer is that radiation does not require a medium as a carrier of energy. Also, for conductive and convective modes of heat transfer, the energy transfer is a function of the temperature difference between the substances. On the other hand, the radiant energy emitted by a substance is a function of the fourth power of the absolute temperature. Thus, the radiative heat transfer becomes dominant at high temperatures.

Radiation heat transfer is very important in semiconductor applications. Basic models for radiative heat transfer, like surface-to-surface, can be solved fairly easily. However, the more complex problems for semiconductor applications that involve participating media, specular radiation, and thin film growth require much more complex methods such as the Discrete Ordinates Method (DOM) and the Monte Carlo method. CFD-ACE+ supports all of these methods. You can opt for one of the methods based on considerations of computational speed and accuracy. The Radiation Module includes:

Radiation-Applications

Radiation-Features

Radiation-Theory

Radiation-Limitations

Radiation-Implementation

Radiation-Frequently Asked Questions

Radiation-References

Radiation Module Applications

Rapid Thermal Processing and Rapid Thermal Chemical Vapor Deposition are two important applications of radiation heat transfer in semiconductor systems. Radiative heat transfer is used to heat the wafers to enhance deposition rates in many chemical vapor deposition (CVD) systems. The Radiation Module has automotive applications in climate control and underhood cooling.

The Radiation Module can:

Obtain surface temperatures of individual and stacked wafers

Be used as a design tool to evaluate wafer temperatures and prevent damage to the wafers during the manufacturing process

Evaluate the change in growth rates of thin films with and without the effects of radiation heating

184

Radiation Module

Radiation Module Features

The Radiation Module has the following features:

Surface-to-Surface

Discrete Ordinate Method (with gray or non-gray properties)

Three Monte-Carlo Methods

P1 Method

Radiative heat transfer problems can be solved by using one of the above mentioned methods. Even complex problems can be handled using this capability. Heat transfer through translucent solids and interference by thin films are notable examples of the use of this module to simulate complex problems.

Radiation-Theory

Radiation Module Theory-Blackbody Radiation

A blackbody is a perfect emitter and absorber of radiation. Using quantum mechanical arguments, it has been shown by Planck and later verified by experiments that the spectral distribution of emissive power of a blackbody is given by (Azzam and Bashara, 1977):

(6-1)

where:

(6-2)

where:

h = Planck’s constant (6.6260755E-34 Jsec)

k = Boltzmann’s constant (1.380658E-23 J/K).

is the wavelength of radiation, T is the absolute temperature and c is the speed of light. The above equation is independent of the nature of the material emitting radiation. The figure below shows the blackbody emissive power as a function of wavelength for different absolute temperatures. Two important observations can be made from this figure: (1) the energy emitted at all wavelengths increases with temperature; (2) the peak spectral emissive power shifts toward a smaller wavelength as the temperature increases. Wien derived a relationship for wavelength at which maximum emissive power occurs, given by

185


(6-3)

which is called the Wein’s displacement law. Integrating equation 6-1 over all the wavelengths results in the Stefan-Boltzmann law given by

(6-4)

where is the Stefan-Boltzmann constant (5.669e-8 W/m2K4).

Spectral Emissive Power of a Blackbody at Different Temperatures (Siegel and Howell)

The Planck’s spectral distribution gives the maximum intensity of radiation that any body can emit in a vacuum at a given wavelength and temperature. The energy emitted in a wavelength band required for the non-gray calculation is obtained by calculating the area under the Planck’s curve. The fractional energy emitted in a wavelength band can be obtained analytically using a series approximation developed by Chang and Rhee:

186

Radiation Module

(6-5)

where:

(6-6)

Radiation Module Theory-Radiation Properties

All real substances do not absorb or emit as blackbodies. The emissive power of an arbitrary surface at temperature T to the hemispherical region above it is given by:

(6-7)

where is the (hemispherical) emissivity of the surface which varies between 0 and 1. The emissivity is in general a function of the material, condition of the surface (rough or polished), the wavelength of the radiation and the temperature of the surface.

The monochromatic emissive power of a surface is given by:

(6-8)

where is called the monochromatic hemispherical emissivity. The emissivity and monochromatic emissivity are related by the following equation:

(6-9)

When radiation is incident on a surface, some of the energy is absorbed, some of the energy is reflected and some of the energy is transmitted. This behavior is characterized by: absorptivity (), defined as the fraction of incident energy that is absorbed; reflectivity (), defined as the fraction of energy reflected; and transmissibility () defined as the fraction of energy transmitted. Clearly, the sum of these quantities is unity, i.e.

187


(6-10)

For an opaque surface, the transmissivity is 0 and hence

(6-11)

Kirchhoff’s law states that at thermal equilibrium, the emissivity of a surface is equal to the absorptivity, i.e.

(6-12)

Combining the above two equations:

(6-13)

Radiation Module Theory-Radiation Characteristics of Gases

The absorption and emission characteristics of gases depend on the thermodynamic state of the gas. In general gases absorb and emit only in narrow wavelength bands and hence most of the gases are not gray. Fortunately, many gases are relatively transparent to thermal radiation in temperature ranges of common engineering problems, and their presence can be ignored (non-participating media). However, certain gases (combustion products) participate in radiative transport even at relatively low pressures and temperatures. The gases which have these low temperature radiation characteristics are similar, in that the constituent molecules are non-symmetric and polar. These gases include CO2, H2O, CO, SO2 and many hydrocarbons.

Often, the data on radiative property of gases is presented in terms of emittance (g). But the absorption coefficient is needed to solve the radiative transfer equation. To obtain the absorption coefficient from the emittance data, the following formula can be used:

(6-14)

where Lm is the mean beam length which may be calculated (for optically thin gas radiating to its entire boundary) as:

(6-15)

where V is the volume of the enclosure and A is the area of the boundaries.

Radiation Module Theory-Radiative Transfer Equation (RTE)

The integro-differential radiative heat transfer equation for an emitting-absorbing and scattering gray medium can be written as:

188

Radiation Module

(6-16)

where is the direction of propagation of the radiation beam, I is the radiation intensity which is a function of both position (r) and direction (), and are the absorption and scattering coefficients respectively, Ib is the intensity of black body radiation at the temperature of the medium and is the phase function of the energy transfer from the incoming ‘ direction to the outgoing direction . The term on the left hand side represents the gradient of the intensity in the specified direction . The three terms on the right hand side represent the changes in intensity due to absorption and out-scattering, emission and in-scattering respectively. The heat transfer is schematically shown below.

Radiative Heat Transfer in an Emitting-Absorbing and Scattering Gray Medium

The boundary condition for solving the above equation 6-16 may be written as:

(6-17)

where I is the intensity of radiant energy leaving a surface at a boundary location, is the surface emissivity, is the surface reflectivity, and n is the unit normal vector at the boundary location.

Theory-Solution Method

Radiation Module Theory-Solution Method Introduction

A number of numerical techniques are available for solving the radiative transfer equation. The following methods have been implemented in CFD-ACE+ for the solution of the radiative heat-transfer equation. The Solution Method section includes:

189


Solution Method-Surface-to-Surface

Solution Method-Discrete Ordinate Method

Solution Method-Monte-Carlo Method

Solution Method-Monte Carlo Raytracing

Solution Method-Patch Definitions

Solution Method-Radiative Properties

Radiation Module Theory-Solution Method-Surface-to-Surface Method

If the optical thickness of the participating medium is very thin the right hand side of equation 6-16 is zero. The solution technique is very similar to the YIX method (Tan and Howell, 1990).

(6-18)

The integral formulation of the above equation is:

(6-19)

where:

= the blackbody emission power

= the Stefan-Boltzmann constant

qs, = the surface radiation flux

= the radiosity

n and n' are normal at r and r', respectively

The kernel, K , is defined as:

where is the visibility function defined as:

Equation 6-19 can be written as:

190

Radiation Module

(6-20)

where:

=

the location hit by the ray emitted from r in the direction.

The angular integral on the right hand side of equation 6-20 can be replaced by numerical quadrature of the form:

(6-21)

where is the angle between the normal and and M is the number of angular integration points. Then the discrete form of equation 6-20 is:

(6-22)

where is the hit point of the ray emitted from inj

Solution Method-Discrete Ordinate Method

Radiation Module Theory-Solution Method-Discrete Ordinate Method-Introduction

The Discrete Ordinate Method (Fiveland, 1988) has been implemented in CFD-ACE+ because it is well suited for accurately predicting radiative heat transfer for most engineering applications.

In the discrete ordinate method, equation 6-16 and equation 6-17 are replaced by a discrete set of equations for a finite (specified) number of ordinate directions. The integral terms on the right hand side of equation 6-16 is approximated by a summation over each ordinate. The discrete-ordinate equations may then be written as:

(6-23)

For the gray model, the subscript should be dropped from the above equation and F becomes unity. In the non-gray model, the radiative properties are assumed to be functions of wavelength

191


only. For strongly participating media such as combustion products, in addition to the wavelength dependence, the radiative properties are functions of local temperature, pressure, and composition of the gas. Hence, the radiative properties need to be calculated using either narrow-band or wide-band models and they should be provided as input to this model.

This section includes:

Discrete Ordinate Method-Wall Boundary

Discrete Ordinate Method-Symmetry Boundary (specular reflection)

Discrete Ordinate Method-Inlet and Exit Boundary

Discrete Ordinate Method-Conjugate Heat Transfer

Discrete Ordinate Method-AAQ Model

Radiation Module Theory-Solution Method-Discrete Ordinate Method-Wall Boundary

For an adiabatic wall condition, the wall temperature is calculated by balancing the radiative and conductive heat flux.

(6-24)

Radiation Module Theory-Solution Method-Discrete Ordinate Method-Symmetry Boundary (specular reflection)

(6-25)

Radiation Module Theory-Solution Method-Discrete Ordinate Method-Inlet and Exit Boundary

(6-26)

where Ti is the cell center temperature adjacent to the boundary.

In the above equations, m and m' denote the outgoing and incoming directions, respectively. For a direction m, wm represents the associated weight while , , and represent the direction cosines corresponding to the x, y and z coordinates respectively. Equation 6-23 represents M coupled partial differential equations for M intensities, Im.

In CFD-ACE+, the S4 approximation, which considers 12 ordinate directions in two dimensions (24 in three dimensions) was chosen. The selection of ordinate directions is not arbitrary but must satisfy the symmetry and moment invariance constraints.

192

Radiation Module

The in-scattering term on the right hand side of equation 6-23 contains the phase function which is dependent on the medium. In CFD-ACE+, the medium is assumed to be linearly anisotropic for which the phase function may be written as:

(6-27)

where a o is an asymmetry factor that lies between -1 and 1. The values -1, 0, 1 denote backward, isotropic and forward scattering, respectively. In CFD-ACE+, the in-scattering term is evaluated explicitly using the previous iteration values and hence the discrete-ordinate equations are de-coupled and the equations are solved sequentially.

Equation 6-23 is numerically integrated over each control volume of the flow domain for each ordinate direction ‘m’ using techniques. (See Appendix C for details.)

Under conditions of local thermodynamic equilibrium, the net radiative heat source in a computational cell is the difference between the energy absorbed and the energy emitted, given by:

(6-28)

where is the volume of the cell. This source term is added to the discretized fluid enthalpy equation. This source term will be zero for a non-participating medium = 0.

Radiation Module Theory-Solution Method-Discrete Ordinate Method-Conjugate Heat Transfer

For conjugate heat transfer including radiation, the temperature of the gas-solid interface is required to calculate the radiation intensity boundary condition. At the interface, the heat flux from both sides must be continuous. i.e.

(6-29)

where Kg and Ks are thermal conductivities of the gas and solid, respectively. In discretized form, the above equation may be written (for orthogonal grids) in terms of the interface temperature (Ti) as:

(6-30)

where:

(6-31)

and Qrad is the net radiative heat flux at the interface which is the difference between the radiative energy absorbed and the energy emitted at the interface. The emission term in the above

193


equation is linearized to obtain a semi-implicit solution for the interface temperature. The net gain/loss of heat due to absorption/emission is added as a source term to the energy equation on both sides of the interface as:

(6-32)

For body-fitted-coordinate (BFC) grids, the approach is similar but includes non-orthogonal cross-terms. For turbulent flows, the thermal conductivity of the gas in the above equation is replaced by an effective thermal conductivity. The effective thermal conductivity is evaluated from wall functions for turbulent momentum and thermal boundary layers.

At the interface between a transparent solid and gas, the above source terms are not included because it is accounted by solving the discrete-ordinate equations in the transparent solid using the appropriate absorption coefficient.

Radiation Module Theory-Solution Method-Discrete Ordinate Method-AAQ Model

The AAQ model provides a more conservative formulation in calculating the incidence radiative flux on a wall for body-fitted coordinate geometries. In the discrete ordinate approach, intensities are calculated along pre-selected directions. Each selected direction accounts for radiation within a solid angle of Wi which is the weighting factor for direction i.

When calculating the incident flux on a wall, even though the representative ray may be along the incoming direction, some of the rays within the included solid angle may not be along the incoming direction. To account for this, a correction factor is applied to the calculated incident radiation flux. This correction factor is calculated based on the fact that the summation of the solid

angles associated with the incoming rays must add up to

Solution Method-Monte Carlo Method

Radiation Module Theory-Solution Method-Monte Carlo Method

The Monte Carlo Method is considered one of the most accurate methods for the calculation of radiative heat transfer. This is because of its ability to treat all directions of radiative transfer in a continuous fashion (rather than along discrete directions, as in the Discrete Ordinates Method), and its ability to account for strong oscillations in the spectral radiative properties. In addition, it is the only method that can treat non-diffuse reflection from walls.

Although the Monte Carlo Method can be used to predict radiative transfer in any scenario, this particular model was developed with the semiconductor material processing industry in mind. Thus, its strength is best realized for Rapid Thermal Processing and Rapid Thermal Chemical Vapor Deposition applications, and in general, for simulation of radiative heat transfer in semiconductor processing applications.

In general, the radiative transfer equation can be solved using the Monte Carlo approach by tracing photon bundles (or rays) through discrete control volumes, and by accounting for the various events (absorption, emission and scattering) occurring within each control volume. Such volumetric raytracing, however, is prohibitively expensive. Furthermore, thin films cannot be modeled using this approach because thin films grown by CVD are often a few microns thick, while the reactor dimensions are in the order of tens of centimeters. Typically, CVD reactors operate at low pressure. Quite often, more than 80% of the gas mixture in the reactor is comprised of an inert gas such as argon. Under these circumstances, it is justifiable to assume

194

Radiation Module

that the gas within the reactor is non-participating. In the absence of participating gases, the energy of a ray remains unchanged as it passes through the gas and therefore, the solution to the radiative transport equation reduces to energy exchange between surfaces. Participating solids can be treated by invoking the McMahon approximation, and by lumping the effect of the solid volumes to the surface (i.e., boundary conditions). The exchange of energy between the various radiatively active surfaces (so called patches) may be described by the following equation:

(6-32)

where:

= Heat Flux (w)

= Heat Flux density (w/m2)

= Area (m2)

= Kronecker delta

= Emissivity of Patch j

=

Radiation exchange matrix (fraction of radiation emitted by patch i and absorbed by patch j

= Stefan-Boltzmann constant (5.669 x 10-8 W/m2k4)

= Average temperature of patch j (K)

Radiation Module Theory-Solution Method-Monte Carlo Raytracing

In a Monte Carlo raytracing scheme, rays are emitted from a surface and traced until they are absorbed by the same surface or any other surface. The emission, absorption, reflection or refraction of the ray depends on the radiative properties of the surfaces on which the ray strikes, and certain stochastic relations (see (Modest, 1993) and (Mazunder and Kersch, 2000)).

Radiation Module Theory-Solution Method-Patch Definitions

In principle, it is possible to define each boundary cell face as a patch. This, however, leads to two problems. The end result of Monte Carlo raytracing is the radiation exchange matrix Rij. This matrix has a size Np X Np, which implies that for a simple problem involving 1000 boundary cell faces, one would need to store 106 real numbers. This is prohibitive, and not feasible for practical problems. The second problem is that some boundary cell faces may be extremely small. This will results in collection of very few rays, if at all. The solution for these cells will, thus, be very poor statistically. In order to circumvent this problem, boundary cell faces are grouped in to so-called patches. A patch is a group of faces which have identical radiative properties, have the same temperature, and can only consist of cell faces that are adjacent to each other. From this definition, the following conclusion may be drawn, and should be kept in mind while defining patches.

195


1. Two cell faces cannot be grouped together to form a patch if their radiative properties are not the same, even if all the other criteria are met.

2. The cells that are grouped together to form a patch must be adjacent to each other. The assumption is that the radiative heat flux on the patch is uniform on each of the cell faces belonging to the patch, its value being the average flux. If two cell faces are not adjacent and at completely different geometric locations, the radiative fluxes on them cannot be equal because of different view factors associated with them, and the assumption of uniform heat flux on them is, by definition, violated.

3. The third criteria is that the temperature in each of the cell faces belong to a given patch must be the same. Isothermal walls fit this criteria without any problem. Other surfaces (prescribed flux or conjugate surfaces) will not necessarily fit this criteria. The criteria can be relaxed by assuming that the temperature on the cell faces is more or less uniform, and is represented by an average temperature.

In the Monte Carlo Model, patches are created by specifying how many sub-patches each boundary face should be divided into. The sub-patches are then created so as to be at approximately equal number of cell faces. For example, if a user is performing MC calculations for radiative heating of a wafer and is interested in heat flux distributions on the wafer, the wafer surface should be divided into sub-patches sufficient to resolve the expected variation in heatflux.

The larger the area of a patch, the more rays it will collect, and the better will be the statistical accuracy of the solution. Thus, while specifying more patches will resolve variations in heat flux, it may result in poor solution accuracy unless the number of rays (and computational cost) is increased as well. Successful use of the Monte Carlo module involves a careful compromise between accuracy and computational effort.

Radiation Module Theory-Solution Method-Radiative Properties

Since the Monte Carlo model performs high accuracy spectral radiation calculations, high resolution spectral radiative properties are required to exploit the strength of the model. Such property data is not always easy for the user to provide. To circumvent this problem, CFD-ACE+ computes spectral radiative properties of commonly prevalent materials from first principles (using the theory of geometrical optics and electromagnetic radiation).

These computations employ the complex refractive index of the material in question. These optical properties are stored in a database. Available materials are Silicon, Silicon Dioxide (common window glass), Tungsten, and Liquid Water. In addition, for gray opaque surfaces, you can specify the surface emissivity directly. For details on calculation of radiative properties of surfaces, see (Azzam and Bashara, 1977) in References.

For optical properties of materials, see (Palik, 1985) in References.

Radiation Module Theory-Solution Method-P1 Method

In this method, instead of directional discretization, it is assumed that the intensity can be expanded as an infinite series of Legendre polynomials of increasing order. The idea is derived from the fact that solution of an eigenvalue problem in spherical coordinates results in Legendre polynomials as the eigenfunctions. The series, when substituted into the RTE and manipulated, results in a set of coupled diffusion-like equations. When only the leading term in the series is retained (i.e., N = 1 in the expansion), the result is a single Helmholtz equation, the equation, which is written as:

196

Radiation Module

(1)

subject to the boundary condition:

(2)

where is the emissivity of the boundary surface having surface normal . The quantity

is also known as the spectral extinction coefficient. The quantity is known

as the spectral integrated intensity or incident radiation, and is defined as .

is the so-called blackbody emissive power. In deriving equation 1 from the

governing RTE, isotropic scattering has been assumed. Equation 1 is solved in CFD-ACE+ using the standard finite-volume technique. The boundary condition shown in equation 3, which is of the third kind, is implemented in a manner similar to Newton cooling boundary conditions in the heat module.

Coupling to Overall Energy Equation

The divergence of the radiative heat flux appears as a sink in the overall energy transport equation. Once has been obtained by solving equation 1, the divergence in the radiative heat flux can be computed using the following relation:

(3)

where is the radiative heat flux vector. Of practical interest is also the radiative heat flux normal to any surface. This may be written as

(4)

Treatment of Non-Gray Radiation: The Stepwise Gray Model

Equations 3 and 4 clearly suggest that calculation of the radiative heat flux and the source in the energy equation requires solution of the RTE for all wavelengths followed by a spectral integration.

In the case where radiation transport is assumed to be wavelength independent (or GRAY),

equation 1 is solved only once, in which is replaced by .

In the NON-GRAY case, special treatment of equation 1 is necessary. Integration of equation 1 over the entire spectrum yields

197


(5)

In the step-wise gray model, the spectrum is first split into discrete spectral intervals termed bands. Within each band, , the radiative properties (i.e., , , ) are assumed to be constant. Under this approximation, equation 5 may be re-written as a set of gray equations:

(6)

where and are the extinction and absorption coefficients of the -th band, respectively. The total number of bands is denoted by . The quantities and represent the incident radiation and blackbody emissive powers within the -th band, respectively. Following a similar procedure, equations 3 and 4 may be written in discrete form as:

(7)

and

(8)

The equations represented by equation 6 are solved in CFD-ACE+ for the non-gray case, and equations 7 and 9 are finally used to compute the net radiative heat fluxes.

Radiation Module Limitations

The Surface-to-Surface model does not account for any participating medium, hence radiation through semi-transparent solids cannot be handled. It does not work with cyclic boundary conditions.

The Discrete Ordinate Method has problems with specular radiation. The ordinate set implemented in the Radiation module is symmetric only about the x, y and z coordinate axes. Therefore, the specular reflection boundary condition (used for symmetry) is accurately imposed only for boundaries that are aligned with the coordinate axes. It is not very accurate for optically thin media.

The spectral distribution is subdivided into a finite number of bands within which the properties are assumed uniform. In actuality these properties are not uniform and this can lead to inaccuracies. The radiative properties are also highly dependent on the wavelength of the light. Since uniform properties are assumed in a spectral band this can also lead to inaccuracies in the solution.

The Monte Carlo Method cannot treat radiative transfer through participating gases. It can only treat participation in solids. Arbitrary interfaces work with Monte Carlo radiation only if they are used at fluid-fluid boundaries. It does not support interfaces between two semitransparent solids. Also, it is slow compared with the other radiation models in CFD-ACE+ and should be used only for cases where its features are necessary. The Monte Carlo Method cannot be used with thin walls.

198

Radiation Module

Radiation-Implementation

Radiation Module Implementation-Introduction

The Implementation section describes how to setup a model for simulation using the Radiation Module. The Implementation section includes:


Model Setup and Solution - Describes the Radiation Module related inputs to the CFD-ACE-Solver


Radiation Module Implementation-Grid Generation

The Radiation Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). Furthermore all grid cell types are supported (quad, tri, hex, tet, prism, poly).



Radiation Module Implementation-Model Setup and Solution-Introduction

CFD-ACE+ provides the inputs required for the Radiation Module. In addition, a <modelname>.PATCH file is required to define the radiative properties of the various patches. Model setup and solution requires data for the following panels:

MC Model Requirements

Problem Type

Model Options

Data

Volume Conditions

Boundary Conditions

Initial Conditions

Solver Control

Output

Radiation Module Implementation-Model Setup and Solution-MC Model Requirements

The Monte Carlo model computes radiative properties from first principles. In addition, it also allows surfaces of many different reflection characteristics (i.e. diffuse, specular, partially specular). It can also account for coatings on surfaces. Thus, specifying radiative properties in this model is not a simple matter of specifying emissivities and reflectivities.

Patch File

199


In the current setup, we employ a simple ASCII file to setup and input these properties. The file must be named <modelname>.PATCH, and must reside in the working directory. See What do the contents of a Patch File look like?.

Optical Database File

The substrate material name is used to fetch its optical properties. The properties stored in the optical database are used to compute the complex refractive indices of materials as a function of its temperature. See What is the Optical Database File?

Radiation Module Implementation-Model Setup and Solution-Problem Type


Select Radiation to activate the Radiation Module. The Radiation Module can work with any of the other modules in CFD-ACE+.

The Heat Transfer Module is required to be activated whenever the Radiation Module is in use. See Heat Transfer Module for details about the Heat Transfer Module.


Radiation Module Implementation-Model Setup and Solution-Model Options-Introduction

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. The Model Options section includes:


Model Options-Radiation (Rad)

Model Options-Model Selection

Radiation Module Implementation-Model Setup and Solution-Model Options-Shared

There are no settings under the Share tab that affect the Radiation Module. See Model Options for details.

Radiation Module Implementation-Model Setup and Solution-Model Options-Radiation(Rad)

All of the model options for the Radiation Module are located under the Radiation (Rad) tab.

Model Options Panel in Radiation Mode

200

Radiation Module

Radiation Module Implementation-Model Setup and Solution-Model Options-Model Selection

In the pull-down menu, there are three options for you to choose a desired radiation model: Discrete Ordinate Method (default), Surface to Surface, and Monte Carlo.

When you choose the Monte Carlo method, the panel changes as shown below. The number of rays that you want to trace must be specified. The default is set at 100000. The accuracy of the Monte Carlo solution is directly related to the number of rays you trace. A good rule of thumb is to use 1000 rays per patch.

Model Options Panel in Monte Carlo Radiation Mode

The Monte Carlo raytracing, being an expensive calculation procedure, cannot be performed for every iteration of the energy equation. Instead it is performed every hundred or so iterations. For example, if you have set up a case for 500 iterations, it may be enough to perform 5 MC updates during the whole run. This means that the Solution Skipping Frequency will be 100. This should be prescribed in the next panel. In some cases, only one MC calculation may be sufficient. An example would be a case where all boundaries are isothermal, and you are interested in the radiative heat flux on the various surfaces. For transient problems, performing MC updates once may also be a good option.

In CFD-ACE+ MC calculations are performed using two kinds of statistical formulations. The Pseudo MC approach employs pseudo random numbers from a uniform deviate, while the Quasi MC approach uses numbers drawn from the Halton sequence. The Quasi MC approach gives slightly more accurate solutions for fewer number of rays, and is therefore, the default option. However, it has some limitations -- it does not work well for cases involving a number of highly reflective surfaces.

Model Setup and Solution-Data

Radiation Module Implementation-Model Setup and Solution-Data-Introduction

When you select the Discrete Ordinate Method (DOM) or the Surface to Surface model, you must specify the related radiative property data. Click the Define Radiation Model Data button or select Radiation Models from the Models menu. The Radiation Model window appears.

201


Radiation Model Window

The radiation model window has tabs for each of the settings:

Wavelengths

Emissivity settings

Transmissivity settings

Absorption Coefficient settings

Radiation Sources

The Model Setup and Solution Data Section includes:

Data-Discrete Ordinate method (DOM)

Data-Surface-to-Surface Method (STS)

Data-Wavelengths

Data-Emissivity Sets

Data-Absorption Coefficient Sets

Data-Radiation Sources

Radiation Module Implementation-Model Setup and Solution-Data-Discrete Ordinate Method (DOM)

If you select the Discrete Ordinate Method, the Radiation Model window appears.

202

Radiation Module

Radiation Model Settings for DOM Method

You will be able to choose whether you want to use the DOM method in the gray (wavelength independent properties) or non-gray (wavelength dependent properties) mode. If you activate Non-Gray, you will be required to input the number of wavelength bands to be used for radiative property specification. See Wavelengths for details on how to define the wavelength bands.

You will also be able to choose how many unique emissivity sets and absorption coefficient sets to use for radiative property specification. Usually each unique surface in the simulation will require a unique emissivity set and each unique solid or fluid volume will require a unique absorption coefficient set. See Emissivity Sets and Absorption Coefficient Sets for details on entering these radiative properties.

Radiation Module Implementation-Model Setup and Solution-Data-Surface-to-Surface Method (STS)

If you select the Surface-to-Surface method, the Radiation Model window appears.

203


Radiation Model Window for STS Method

You will be able to choose the accuracy level of the method. The choices are; Low, Moderate, High, and Extremely High. The default method is Moderate and is usually acceptable for most simulations.

Each higher level of accuracy adds almost an order-of-magnitude to the computational time, while the actual increase in solution accuracy may only be 10%. For this reason you should refrain from increasing the accuracy setting unless absolutely necessary.

The Sub-iteration setting enables you to determine how many times the STS method is called for each solver iteration. The default setting is 1 and is usually sufficient. In some cases, increasing this value can increase the overall convergence rate of the simulation but it will not change the solution.

The Environment Temperature setting is used if any of the surfaces can see outside of the computational domain. This may happen if the domain is not closed, or some surfaces are transparent, or some surface have their surface normal pointing outward. In any case, the surfaces that can see outside of the computational domain will be exchanging radiative heat transfer with a black body at the environment temperature. The default value is 300 K. If you want no radiative exchange with the outside world then you can set the environment temperature to 0 K.

The Radiation Sources feature enables you to model point radiation sources. Once this feature has been activated, you will be able to specify the number of radiation sources that are present. A radiation source can be used to model solar radiation. See Radiation Sources for details on inputting the radiation source parameters.

You can also choose how many unique emissivity sets and transmissivity sets to use for radiative property specification. Usually each unique surface in the simulation will require a unique

204

Radiation Module

emissivity set and a unique transmissivity set. See Emissivity Sets and Radiation Sources for details on entering these radiative properties.

Radiation Module Implementation-Model Setup and Solution-Data-Wavelengths

If the Discrete Ordinate Method (DOM) has been selected and non-gray radiative properties has been activated (see Discrete Ordinate Method), then the Wavelengths tab will become active.

Radiation Wavelengths Settings

The Wavelengths panel lets you define the wavelength bands (in m) that will be used for setting non-gray radiative properties (see Emissivity Sets and Absorption Coefficient Sets for details on the how to specify these properties). You will seen n+1 inputs, where n is the number of bands (n) that you specified on the Model page. The first entry is always 0 microns, and the last entry is always infinity. You define the bands by inputting the cutoff between each band. For example, the inputs shown in the figure above will produce the following three bands:

Note: The number of bands (n) must represent the lowest common denominator of the non-gray radiative properties needed for your particular simulation. This includes emissivity and absorption coefficient properties. For example if you have several materials in your model and their wavelength dependent emissivity and absorption coefficient values are as shown below you will need four bands (even though no single wavelength dependent property requires more than three bands).

205


Example of Non-Gray Property Banding

Data-Emissivity Sets

Radiation Module Implementation-Model Setup and Solution-Data-Emissivity Sets-Introduction

Emissivity data is required for both the DOM and STS methods. The approach is slightly different depending on whether the gray or non-gray property option is selected. The Data Emissivity Sets section includes:

Emissivity Sets-DOM (gray) and STS Methods

Emissivity Sets-DOM (non-gray) Method

Radiation Module Implementation-Model Setup and Solution-Data-Emissivity Sets-DOM (gray) and STS Methods

If you have not activated the non-gray option, the panel will appear for the STS method and for the DOM method as:

206

Radiation Module

Radiation Window-Emissivity Settings for STS and DOM (gray) Methods

The Emissivity Set field allows you to select which emissivity set’s properties are currently displayed in the panel. The total number of emissivity sets was determined on the Model page (see Discrete Ordinate Method (DOM)).

For each emissivity set, you may give a name and a value. The name will be used later when assigning radiative properties to boundary conditions. See Boundary Conditions for details. The value is the total hemispherical emissivity and must range between 0 and 1. Since this panel is for gray property settings the emissivity value will be the same for all wavelengths (i.e., Band 1 = 0 - m).

Use material names for the set names. This will make it much easier to determine which set to apply to each boundary condition.

Radiation Module Implementation-Model Setup and Solution-Data-Emissivity Sets-DOM (non-gray) Method

If you are using the non-gray DOM method, the emissivity panel will look similar to that shown.

207


Radiation Model Window-Emissivity Settings for DOM (non-gray) Method

The Emissivity Set field enables you to select which emissivity set’s properties are currently displayed in the panel. The total number of emissivity sets was determined on the Model page (see Discrete Ordinate Method (DOM)).

For each emissivity set, you may give a name and wavelength band dependent values. The name will be used later when assigning radiative properties to boundary conditions (see Boundary Conditions). The value is the total hemispherical emissivity and must range between 0 and 1.

Since this panel is for non-gray property settings the emissivity value may be different for each wavelength band. The total number of bands was determined on the Model page (see Discrete Ordinate Method (DOM)) and each band’s wavelength range was determined on the Wavelengths page (see Wavelengths).

Use material names for the set names. This will make it much easier to determine which set to apply to each boundary condition.

Data-Absorption Coefficient Sets

Radiation Module Implementation-Model Setup and Solution-Data-Absorption Coefficient Sets-Introduction

Absorption Coefficient data is required for both the DOM and STS methods. For the STS method, absorption coefficient is only used as a switch to determine if a volume condition is opaque or transparent. For the DOM method, the absorption coefficient is used to determine if a volume condition is opaque, transparent, or semi-transparent. The approach is slightly different

208

Radiation Module

depending on whether the gray or non-gray property option has been selected. The Absorption Coefficients section includes:

Absorption Coefficients-STS and DOM (gray) Method

Absorption Coefficients-DOM (non-gray) Method

Radiation Module Implementation-Model Setup and Solution-Data-Absorption Coefficient Sets-STS and DOM (gray)

The Radiation Model-Absorption Coefficient window appears for the STS method and for the DOM method if the non-gray option is not activated.

Absorption Coefficient Settings for the DOM (gray) Method

The Absorption Coefficient Set field allows you to select which absorption coefficient set’s properties are currently displayed in the panel. The total number of absorption coefficient sets was determined on the Model page (see Discrete Ordinate Method (DOM)).

For each absorption coefficient set, you may give a name and a value. The name will be used later when assigning radiative properties to volume conditions (see Volume Conditions). For transparent media the value will be 0, for semi-transparent media the value will be greater than 0, and for opaque media the value should be set to -1. Since this panel is for gray property settings the absorption coefficient value will be the same for all wavelengths (i.e., Band 1 = 0 - m).

If the STS method is being used, then the only valid absorption coefficient settings are -1 (opaque) and 0 (transparent).

Use material names for the set names. This will make it much easier to determine which set to apply to each volume condition.

209


Radiation Module Implementation-Model Setup and Solution-Data-Absorption Coefficient Sets-DOM (non-gray)

The Radiation Model-Absorption Coefficient window appears as below, if the non-gray DOM method is activated.

Radiation Model Widow-Absorption Coefficient Settings for DOM (non-gray) Method

The Absorption Coefficient Set field enables you to select which absorption coefficient set’s properties are currently displayed in the panel. The total number of absorption coefficient sets was determined on the Model page (see Discrete Ordinate Method (DOM) ).

For each absorption coefficient set, you may give a name and wavelength band dependent values. The name will be used later when assigning radiative properties to volume conditions (see Volume Conditions ). For transparent media the value will be 0, for semi-transparent media the value will be greater than 0, and for opaque media the value should be set to -1.

Since this panel is for non-gray property settings the absorption coefficient value may be different for each wavelength band. The total number of bands was determined on the Model page (see Discrete Ordinate Method (DOM)) and each band's wavelength range was determined on the Wavelengths page (see Wavelengths).

Use material names for the set names. This will make it much easier to determine which set to apply to each volume condition.

Radiation Module Implementation-Model Setup and Solution-Data-Radiation Sources

You can use Radiation Source by the STS method to model the effects of any point source radiation. This is most often used to model solar radiation effects. If you activated Radiation

210

Radiation Module

Sources in the Model panel (see Discrete Ordinate Method (DOM)), you will be able to press the Radiation Sources tab to see a panel similar to that below.

Radiation Model Window-Radiation Sources Tab

The Radiation Sources field enables you to select the radiation source’s properties that appear in the panel. The total number of radiation sources was determined on the Model panel (see Discrete Ordinate Method (DOM)).

For each radiation source, you need to give a direction and a intensity value. The direction is specified by supplying a normal vector which states the direction from the radiation source to the model origin. The radiation source itself is assumed to be located an infinite distance away from the origin. The intensity value (W/m2) determines the intensity of the radiation source. (For solar radiation the intensity is usually on the order of 700-800 W/m2).

The only way an external radiation source will see any of your computational boundaries is by one of three situations; a surface boundary condition has it’s normal reversed to point outside of the domain, a surface boundary condition is transparent, or the computational grid system is not closed.

Radiation Module Implementation-Model Setup and Solution-Volume Conditions-Introduction

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. See Control Panel-Volume Conditions for details. Before any property values can be assigned, one or more volume condition entities must be made active by picking valid entities from either the Viewer Window or the VC Explorer.

For MC radiation method, there is no need to specify any volume condition. But for DOM and STS models, there are two volume condition settings available; Absorption Coefficient Set and Emissivity Set.

211


Absorption Coefficient Set

Once you have selected one or more volume conditions, you may select a previously defined absorption coefficient set (see Absorption Coefficient Sets) to use for that volume condition. This information is needed for both the Surface-to-Surface method and the Discrete Ordinate Method. For the Surface-to-Surface method, the absorption coefficient data is only used to determine if the material is opaque (absorption coefficient = -1) or transparent (absorption coefficient = 0).

Emissivity Set

You may also select a previously defined emissivity set (see Emissivity Sets) to use for the volume conditions. The emissivity set is only applied to opaque/transparent or opaque/semi-transparent volume condition interfaces. At these locations the emissivity set from the opaque side will be used to determine the emissivity of the opaque surface.


Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Introduction

Click the Boundary Conditions [BC] tab to see the Boundary Condition Panel. See Control Panel-Boundary Conditions for details. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or the BC Explorer.

All of the general boundary conditions for the Radiation Module are located under the Radiation (Rad) tab and can be reached when the boundary condition setting mode is set to General. Each boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). See BC Type for details on setting boundary condition types. This section will discuss the implementation of each type with respect to the Radiation Module.

The Radiation Module handles boundary conditions slightly differently than most of the other modules. Because radiation is a ray based phenomena, a radiation boundary condition must be given for all of the computational boundaries in the model. This means that even inlets and outlets must have radiation boundary conditions.

The Boundary Conditions section includes:

Boundary Conditions-Emissivity Set

Boundary Conditions-Transmissivity Set

Boundary Conditions-Surface Normal

Boundary Conditions-Farfield

Boundary Conditions-Surface Property

Boundary Conditions-Radiation Temperature

Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Emissivity Set

The emissivity set setting enables you to specify which previously defined emissivity set (see Emissivity Sets) should be used for the computational surface. This setting is available for both the STS and DOM methods and is also available for all boundary conditions (except for symmetry and interface).

Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Transmissivity Set

212

Radiation Module

The transmissivity set setting enables you to specify which previously defined transmissivity set (see Radiation Sources) should be used for the computational surface. This setting is only used by the STS method.

Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Surface Normal

The surface normal setting is used only by the STS method. If the surface normal is set to Inward, then the computational boundary is exchanging radiative information with the interior (volume) of the computational system. If the surface normal is set to Outward, then the computational boundary is exchanging radiative information with the external environment. The temperature for the external environment can be specified in the Radiation Model panel (see Discrete Ordinate Method (DOM)).

Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Farfield

This type of boundary condition is required only when the MC radiation method is selected and is to allow external radiation, which is completely decoupled from conduction boundary conditions. For example, it we wanted to simulate sunlight coming in through the roof, we can do that best by using a farfield boundary condition. This would bring in radiation corresponding to the temperature of the sun, but conduction fluxes would be based on a much lower (room) temperature. If a farfield BC is not used in this case, one would have to prescribe a very high temperature on the roof itself to simulate radiation by the sun. This will result is tremendous conduction fluxes, resulting in undesirable heating of the room.

Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Surface Property

This type of boundary condition is available only to MC method. You are required to specify a string, representing a surface property type (or patch type). The string is then used to fetch the actual radiative properties from a patch file, which must be named <modelname>.PATCH, and must reside in the working directory. The contents of the patch file are described in MC Model Requirements.

Radiation Module Implementation-Model Setup and Solution-Boundary Conditions-Radiation Temperature

Radiation Temperature represents the radiation temperature of the boundary patch in question. It is used only when MC radiation model is selected and it is necessary for two reasons. First, it serves as the initial guess for property calculations (which are temperature dependent) and also serves as the initial guess for the first MC calculation. After subsequent MC calculations, this temperature is replaced by the average patch temperature that is calculated by solution of the overall energy equation. The radiation temperature is also necessary for farfield boundary conditions, in which case, it serves as the temperature of the farfield source, and is never updated.

If a wall or rotating wall boundary condition has it’s heat transfer subtype set to adiabatic, then the solver will force the net heat flux (conduction+convection+radiation) to zero.

On a symmetry boundary condition all heat flux values (radiation, conduction, and convection) are forced to zero.

Radiation Module Implementation-Model Setup and Solution-Initial Conditions

213


There are no Initial Condition settings required by the Radiation Module.

Radiation Module Implementation-Model Setup and Solution-Solver Control

Click the Solver Control [SC] tab to see the Solver Control Panel. See Control Panel-Solver Control for details.

The Solver Control page allows access to the various settings that control the numerical aspects of the CFD-ACE-Solver and all of the output options.

There are no general numerical control settings required by the Radiation Module. There are some controls (sub-iterations and accuracy) for the Surface-to-Surface model which are accessible from the Radiation Model panel (see Surface-to-Surface Method (STS) for details.)

Radiation Module Implementation-Model Setup and Solution-Output

There are no settings under the Output tab that affect the Radiation Module. See Control Panel-Output Options for details.

Printed Output

Under the Printed tab, select the printed information to be written to the text based output file (modelname.out). Activation of the Radiation Module allows output of a heat transfer summary in addition to the general printed output options (see Printed Output for details on the general printed output options including boundary condition integral output, diagnostics and monitor point output).

The heat transfer summary will provide a tabulated list of the integrated heat transfer (J/s) through each of the thermal boundary (walls, inlets, outlets, interfaces, etc.). This summary will separate the heat transfer due to radiation and the heat transfer due to conduction/convection.

Graphical Output

Under the Graphics tab, you can select the variables to output to the graphics file (modelname.DTF). These variables will then be available for visualization and analysis in CFD-VIEW. Activation of the Radiation Module allows output of the variables listed.

Radiation Module Related Graphical Output

Variable Units

Radiative Wall Heat Flux

W/m2

Radiation Module Implementation-Post Processing

CFD-VIEW can post-process the solutions. When the Radiation Module is invoked, the temperature field is usually of interest. The temperature field can be visualized with surface contours and analyzed through the use of point and line probes.

The heat transfer summary written to the output file (modelname.out) is often used to determine quantitative results. The heat transfer summary can also be used to judge the convergence of the

214

Radiation Module

simulation. Due to the law of energy conservation, the summation of all heat transfer into and out of the computational domain should be zero (unless heat sources or sinks are present). In the simulation a summation of exactly zero is almost impossible, but you should see a summation that is several orders of magnitude below the total heat transfer into the system.

Radiation Module Frequently Asked Questions

Why do I have to specify absorption coefficient data for the Surface-to-Surface (STS) method?

The absorption coefficient data is the mechanism used to tell the Radiation Module whether a volume condition (fluid or solid) is transparent or opaque. If the Radiation Module sees a volume condition with an absorption coefficient value of -1 then it knows that the volume condition is opaque, likewise, if the value is 0 then it know that the material is transparent.

How do I decide how many patches I must set up for MC method?

The answer to this question is not straightforward. It depends primarily on what you are seeking. In general if you need to resolve surface radiative fluxes, you need more patches on that surface. This is a perfect analogy with using finer grids in regions where you want to resolve gradients/scales in traditional CFD analysis. Remember, the larger the patches, the better the statistical accuracy. So, do not use a large number of small patches, if they are not necessary for what you are seeking. For example, if I am interested in the average transient response of a wafer as it is heated, I will choose to make the whole wafer surface a single patch. On the other hand, if I am interested in the center to edge nonuniformity in temperature of the wafer, I will set a large number of patches on the wafer surface.

How many rays do I use in MC radiation model?

The number of rays to be used is determined by how many patches you have, whether you have surfaces with strong non-gray properties, how many of these you have, the dimensionality of the problem (2D/3D), and various other secondary factors. In general, for 2D 100,000 is a good number and for 3D 1 million is reasonable, although in some cases larger numbers may be necessary. At the higher end, the gain you will have by increasing number of rays will be very minimal, i.e., when you increase number of rays from 10000 to 20000, your solution accuracy may improve significantly, but when you increase by the same number from 1000000 to 1010000, the solution may hardly change.

How many Monte Carlo updates should I use and how frequently should I use Monte Carlo updates?

This depends on the problem at hand. If there are few non-isothermal patches, a couples of updates my be sufficient. Typically, out experience has shown that a frequency of 100-200 iterations works best for steady state runs. For transient runs, one or two updates in each time-step is typically sufficient, unless your time-steps are very large.

What do the contents of a Patch File look like?

In the current setup, we employ a simple ASCII file to setup and input these properties. The file must be named <modelname>.PATCH, and must reside in the working directory. Its contents look as follows:

WALL Name of surface property type

DUMMY Name of surface substrate material

SPECULAR Reflection Characteristic

GRAY Gray/Non-gray

215


0.15 Emissivity

--------------------------------------- WINDOW Name of surface property type SIO2 Name of surface substrate material DIFFUSE Reflection Characteristic

NONGRAY Gray/Nongray

ST Opaque/Semitransparent (only if Nongray)

UNCOATED Coated/Uncoated (only if Nongray)

This particular example contains two surface property types (or patch types), and explanations are provided for each of the strings/numbers. In general the following guidelines are to be followed while setting up a surface property type (or patch type).

Line 1: Name of Property type. Do not exceed 16 characters for the name. The name typed here must be exactly the same (including cases of individual letters) as the one typed in the Surface Property box under the BC/RAD tab. The string must be continuous (no blank spaces), and can be any character available on a standard computer keyboard.

Line 2: Substrate Material Name. If the surface is gray and opaque, any name can be used (for example DUMMY is used in the example above) because it is never used. If the surface is Nongray/Semitransparent, this line must be filled with an appropriate material name. The optical properties corresponding to this material name will then be fetched from ESI CFDs optical database.

Line 3: Surface Reflection Characteristics. The options are SPECULAR, DIFFUSE and PARTIALLY_SPECULAR. If PARTIALLY_SPECULAR is chosen, the next line must have a real number representing the degree of specularity. For the other two options no other input is necessary.

Line 4 (Line 5 if Line 3 has PARTIALLY_SPECULAR): Spectral Characteristic. Options are GRAY and NONGRAY (or NON-GRAY). If GRAY the next line must have a real number, representing the value of surface emissivity. The surface is assumed opaque in this case, as well.

The next few lines are required only for the NONGRAY option.

Line 5 (Line 6 if Line 3 has PARTIALLY_SPECULAR): Transparency Characteristics. This option is only required for Nongray surfaces (for gray surfaces, opacity is assumed). The options are OPAQUE or ST (or SEMI).

Line 6 (Line 7 if Line 3 has PARTIALLY_SPECULAR): Coating Characteristics. Options are COATED or UNCOATED.

The next few lines are required only for COATED.

Line 7 (Line 8 if Line 3 has PARTIALLY_SPECULAR): Number of layers in coating (integer input).

Line 8 (Line 9 if Line 3 has PARTIALLY_SPECULAR): Material name followed by a blank space followed by the layer thickness in meters.eg. for a two-layered material, we have:

SIO2 1.0E-6

SILICON 4.0-7

Any line beginning with a dash (-) may be used as a separator between two surface property definitions. Comments may be added to this line, if desired.

What is the Optical Database File?

216

Radiation Module

As mentioned earlier, the substrate material name is used to fetch its optical properties. The properties stored in the optical database are used to compute the complex refractive indices of materials as a function of its temperature. The refractive index of a material is written as:

(6-34)

where:

= complex refractive index

n = real part of (refractive index)

k = imaginary part of (absorption index)

i =

where n is the real part and k is the imaginary part of the refractive index. These indices are curve-fitted for temperature as follows:

(6-35)

where

n = real part of refractive index

= coefficients for curve-fit

k =non-dimensional temperature, (T - 300)/1000

(6-36)

where

k = imaginary part of refractive index

= coefficients for curve-fit

The optical database contains a file, listing n0 through n3 and k0 through k3 in the following order:

n0 n1 n2 n3 k0 k1 k2 k3

for sixty different wave numbers. The wave numbers are selected according to the formula

(6-37)

where

= central wave number of i-th band (m-1)

C = 94.40608

217


This allows treatment of any radiation phenomenon between 300 and 5000K. The number sixty stems from the fact that properties are not usually available at better resolution, and sixty was deemed an adequate number.

In the event where the substrate material being used is not part of the optical database provided by ESI CFD, the user would be required to input the n0 through k3 values in the form of an optical file. This file must be named <modelname>.OPTIC, and must reside in the working directory. Its format has already been described, and an example is shown below:

1 Number of Materials in File

TUNGSTEN Material Name

0.5 0.5 Irrelevant for ACEU, only for ACE

85.000 0.000 0.00 0.000 0.000E+00 0.999E+020 .000E+00 0.000E+00

n0 through k3 (sixty rows)

When is P1 Radiation appropriate?

218

Radiation Module

219

The P1 model is invalid for non-participating medium. Mathematically, if goes to zero, the left hand side of the P1 equation goes to infinity. Physically, in this model, radiation is modeled as a diffusive process. This diffusion idea is valid only for optically intermediate-thick situations. If the medium is optically very thin or non-participating, transport of radiation is ballistic, and a diffusion model is invalid. CFD-ACE+ has been set up such that the model will also produce results for non-participating medium (instead of giving division by zero!). However, these results may not be always accurate.

The P1 model is known to work best (see discussions in Radiative Heat Transfer, Second Edition, Academic Press, M.F. Modest) in situations where the medium is hot and strongly emitting/absorbing. In situations where the medium is cold and most of the emission is from hot boundaries, the model is not very accurate. Thus, from an application standpoint, the model is expected to be quite accurate for combustion applications, but is not expected to be very accurate for applications in semiconductor material processing, such as rapid thermal processing and rapid thermal chemical vapor deposition.

For cases in which the extinction coefficient, , is either too small or too large, the P1 equation is very stiff, and convergence will be slow, if at all attainable. The model equation has best convergence properties for intermediate optical thickness or extinction coefficient values.

Radiation Module References

Azzam, R., and Bashara, N., Ellipsometry and Polarized Light. New York: Elsevier Noth-Holland, 1977.

Fiveland, W.A., “Three Dimensional Radiative Heat-Transfer solutions by the Discrete Ordinates Method.” Journal of Thermophysics and Heat Transfer. 2.4 (Oct 1988): 209-316.

Mazunder, S., and Kersch, A., “A Fast Monte Carlo Scheme for Thermal Radiation in Semiconductor Processing Applications.” Numerical Heat Transfer 37.B (2000): 185-199.

Modest, M.F., Radiative Heat Transfer. McGraw Hill, 1993.

Palik, E., Handbook of Optical Constants of Solids. New York: Academic Press, 1985.

Siegel R., and Howell, J.R., Thermal Radiation Heat Transfer. 2nd ed. New York: Hemisphere, 1981.

Tan A., Wang, D., Srinivasan, K., and Przekwas, A.J., “Numerical Simulation of Coupled Radiation and Convection for Complex Geometries." AIAA-98-2677, 1988.

Tan, Z., and Howell, J.R., “New Numerical Method for Radiation Heat Transfer in Nonhomogeneous Participating Media.” AIAA J. Thermophys Heat Transfer 4 (1990): 419-424.

Cavitation Module

Cavitation Module Introduction

The Cavitation Module uses the full cavitation model (Singhal et al, 2001) (Athavale et al, 2000) developed by ESI CFD. It allows multi-dimensional simulations of cavitating flows with phase changes in low pressure regions. The model accounts for important effects such as bubble dynamics, turbulence, and the presence and expansion of non-condensable gases in liquid.

In engineering flows, common performance indicators such as suction head, thrust, lift, and drag are all functions of mass flow rate and pressure distribution. In many equipment engineering considerations, increased noise level (due to cavitation) is also an acceptance/rejection criterion.

The Cavitation Module assists in answering three basic questions:

Will cavitation occur in a given design?

If cavitation is unavoidable, can the given design still function properly?

If the given design is unsatisfactory, what are the ways to reduce or eliminate cavitation?

The cavitation model can predict performance parameters such as realistic distributions of pressures, velocities and void fraction (i.e., volume fraction of vapor and non-condensable gases). The Cavitation Module includes:

Cavitation-Applications

Cavitation-Features

Cavitation-Theory

Cavitation-Limitations

Cavitation-Implementation

Cavitation-Frequently Asked Questions

Cavitation-References

Cavitation-Applications

Cavitation Module Applications-Introduction

Cavitation is a common problem for a myriad of engineering devices in which the main working fluid is in a liquid state. Examples include turbo-pumps for rocket propulsion systems, industrial turbo-machinery, hydrofoils, marine propellers, fuel injectors, hydrostatic bearings, shock absorbers, and biomedical devices such as mechanical heart valves. The deleterious effects of cavitation include:

Lowered system performance

Load asymmetry

Erosion and pitting of solid surfaces

Vibration and noise

Reduction of the life of the machine as a whole

There are also some desirable applications of cavitation:

220

Cavitation Module

Washing machines

Surgical procedures using power cutting with ultrasonic energy

Liquid-solid separators

Removal of organic contaminants from water (using cavitation to purify water)

Ultrasonic cleaning

The Cavitation Module Applications section includes:

Cavitation-Automotive/Hydraulic Applications

Cavitation-Turbomachinery Type Problems

Cavitation-Hydrofoil Type Problems

Cavitation Module Applications-Automotive/Hydraulic Applications

Both vane and gerotor oil pumps have been studied with CFD-ACE+. These simulations use the Cavitation Module in conjunction with rotating/deforming grids (Deformation Module) to accurately predict the pressure profiles and mass flow rates through the device.

Vane Oil Pump (4000 RPM) - Cavitation Inside Pumping Pockets

221


Vane Oil Pump (2000 RPM) - Pressure Profiles Without Cavitation Model (left) and With

Cavitation Model (right)

Effect of Cavitation on Volume Flow Rate

The model has also been used to solve for cavitating flow in a gerotor oil pump as shown below.

222

Cavitation Module

Cavitation Inside Pumping Pockets (5000 RPM) of a Gerotor

Cavitation Module Applications-Turbomachinery Problems

Turbomachinery applications also often require the use of the Cavitation Module. The figure below shows results for an axial flow water pump. The second figure shows results for a centrifugal pump.

223


Axial Water Pump Results

224

Cavitation Module

Centrifugal Pumps

Cavitation Module Applications-Hydrofoil Problems

The Cavitation Module has been validated with numerous problems. Shown below are the results of a 2D hydrofoil simulation along with comparison to experimental data.

225


2D Hydrofoil Simulation

Cavitation Module Features

Numerical simulation of cavitation flows poses unique challenges both in modeling the physics and in developing a robust numerical methodology. Computational Fluid Dynamics (CFD) analysis is complicated by the large density changes associated with phase change. For example, the ratio of liquid to vapor densities for water at room temperature is over 40,000. Typical density variations in engineering flows are indicated below.

Density Ratios in Engineering Flows

Flow Type max/min

Buoyant Flows ~ 1

Transonic Flows ~ 2

Supersonic Flows ~ 10

Reacting Flows ~ 20

226

Cavitation Module

Boiling/Condensating Flows ~ 200

Cavitating Flows ~ 10,000

The location, the extent, and the type of cavitation are strongly dependent on the pressure field, which is strongly influenced by geometric detail and the motion of liquid and vapor phases. The cavitation region is also influenced significantly by turbulence and presence of non-condensable gases. The current model does not require an a priori prescription (or assumption) of the location or type of cavitating region. Likewise, the phase change correlations have minimal empiricism; therefore, various flow conditions can be simulated without adjusting any constants or functions.

The present model can be used to simulate flows with:

Large liquid/vapor density ratios (~50,000)

Highly turbulent conditions (due to high pressures, high mass flow rates, or high rotation speeds) and,

Non-condensable gases (e.g., Air, N2, or He) dissolved in or mixed within the liquid.

The model development has been guided by observations from:

A large number of numerical investigations with various cavitation models used or developed at ESI CFD over the past several years;

A large number of experimental investigations and flow visualization studies presented in international conferences and reported in literature.

Cavitation-Theory

Cavitation Module Theory-Introduction

The basic approach is to use standard viscous flow (Navier-Stokes) equations with provisions for variable density and a conventional turbulence model, such as K- model. The mixture density () is a function of vapor mass fraction (f), which is computed by solving a transport equation simultaneously with the mass and momentum conservation equations. The -f relationship is:

(7-1)

where v and l represent the vapor and liquid densities, respectively. In two-phase flows, the use of vapor volume fraction () is also quite common. Therefore, when necessary, it is deduced from f as follows:

(7-2)

The Cavitation Module Theory section includes:

Cavitation-Theory-Vapor Transport Equations

Cavitation-Theory-Effect of Turbulence

Cavitation-Theory-Effect of Non-Condensable Gases

227


Cavitation Module Theory-Vapor Transport Equations

The vapor transport equation is:

(7-3)

where, is the velocity vector, is the effective exchange coefficient, and R is the rate of phase change. The evaporation phase (bubble generation and expansion) is denoted by Re, and the condensation (bubble reduction and collapse) phase is denoted by Rc. The expressions of Re and Rc have been derived from the reduced form of the Rayleigh-Plesset equation (Brennen, 1995), which describes the dynamics of a single bubble in an infinite liquid domain. We have assumed that in most of the engineering flows there are plenty of nuclei for the inception of cavitation. Thus our primary focus has been on proper account of bubble growth and collapse.

The final expressions for Re and Rc are:

(7-4)

(7-5)

where:

C e , C c

= phase change rate coefficients

= surface tension of the saturated liquid

V ch = characteristic velocity

psat = saturation pressure

The phase change rate coefficients are available as user inputs. The default values are Ce = 0.02, and Cc = 0.01. These values have been determined after considerable numerical experimentation over a wide range of flow conditions, for orifice and hydrofoil flows. These values should not be changed without consultation with ESI CFD Technical Support. The only exception is that Ce and Cc may both be set to 0.0 to remove phase change effects from the cavitation model.

As discussed in Reference 1, Vch is approximated by the local turbulence intensity, i.e.,

. For accounting for laminar or low turbulence conditions the minimum value Vch is set to 1.0.

Cavitation Module Theory-Effect of Turbulence

Most engineering flows are turbulent. Furthermore, cavitation normally takes place in the vicinity of low pressure (or locally high velocity) regions, where turbulence effects are quite significant. In particular, turbulent pressure fluctuations have significant effect on cavitating flows. This has been reported by many experimental investigations (Keller, 1997), (Stoffel, 1995), (Bordelon, 1995). The present model accounts for this effect by modifying the phase change rates, as

228

Cavitation Module

described in Singhal, 1995. The magnitude of pressure fluctuations is estimated by using the following empirical correlation from the literature: (Hinze, 1975):

(7-6)

Here and are local density and turbulence kinetic energy.

The saturation pressure (psat) takes the value of a liquid’s saturation pressure ps(T) in the laminar regime, while for turbulent flow:

(7-7)

Cavitation Module Theory-Effect of Non-Condensable Gases

Cavitating flows are sensitive to the presence of non-condensable gases (Reisman et al, 1997), (Watanabe, 1994), (Rood, 1997). In most liquids, there is a small amount of non-condensable gases present in a dissolved and/or mixed state. For example, laboratory water generally has 15 ppm air dissolved in it. In other applications, e.g., marine propellers, etc., this amount may be considerably larger. In high speed rotational flows, there can be small leaks (ingestion of gases).

In the present model, the non-condensable gas is included by prescribing an estimated mass fraction at inlets. This value is held constant throughout the calculation domain. However, the corresponding density (and hence volume fraction) varies significantly with local pressure. The perfect gas law is used to account for the expansion (or compressibility) of gas; i.e.:

(7-8)

Where W is the molecular weight, P is fluid pressure, R is the universal gas constant, and T is fluid temperature. Out of these, W, R, and T are constant prescribed inputs.

The calculation of mixture density (equation 7-1) is modified as:

(7-9)

Where suffixes v, g, and l refer to vapor, gas, and liquid phases. Correspondingly, we have the following expression for the volume fractions of vapor (v) and gas (g):

(7-10)

(7-11)

and:

(7-12)

229


The combined volume fraction of vapor and gas (i.e., v g) is referred to as the Void Fraction ().In practical applications, for qualitative assessment of the extent and location of cavitation, contour maps of void fraction () are important.

Cavitation Module Limitations

The following are a few limitations in the Cavitation Module. These limitations may be removed in future releases of CFD-ACE+.

Fluid Properties

There is no provision for automatically calculating the fluid properties as a function of temperature. Therefore, you must specify the liquid saturation pressure and vapor density, which depend on the operation temperature, and these will remain constant for the simulation. You must also specify the surface tension. All of the default values are for water at 300 K.

Activating Cavitation

Activating the Cavitation Module means that all fluid volume regions in the simulation will use the Cavitation Module.

Isothermal Assumption The Cavitation Module assumes that the flow is isothermal. For this reason, activation of the Heat Transfer Module is not allowed.

Modules Not Supported

The Cavitation Module has not been tested with the following modules:

Heat Transfer

Chemistry

Two-Fluid

Spray

Free Surfaces (VOF)

Cavitation-Implementation

Cavitation Module Implementation-Introduction

The Implementation section describes how to set up a model for simulation using the Cavitation Module. The Implementation section includes:


Model Setup and Solution - Describes the Cavitation Module related inputs to the CFD-ACE-Solver


Cavitation Module Implementation-Grid Generation

230

Cavitation Module

The Cavitation Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). Furthermore all grid cell types are supported (quadrilateral, triangle, hexahedral, tetrahedral, prism, and polyhedral).

The general grid generation concerns apply, i.e., ensure that the grid density is sufficient to resolve solution gradients, minimize skewness in the grid system, and locate computational boundaries in areas where boundary values are well known. Sufficient grid density should be placed in regions where cavitation is expected to occur. A general rule is to have at least five cells in the cavitation region.


Cavitation Module Implementation-Model Setup and Solution-Introduction

CFD-ACE+ provides the inputs required for the Cavitation Module. Model setup and solution requires data for the following panels:

Problem Type

Model Options

Volume Conditions

Boundary Conditions

Initial Conditions

Solver Control

Output

Cavitation Module Implementation-Model Setup and Solution-Problem Type


Select Cavitation to activate the Cavitation Module. The Flow Module is also required when the Cavitation Module is activated. The concurrent use of the Turbulence, Grid Deformation, and/or Stress modules are fully supported.

The Heat Transfer module cannot be activated because of the assumption of isothermal flows. It follows then that the Radiation module is not allowed either.

Use of the Chemistry, Two-Fluid, Spray, or Free Surfaces (VOF) modules have not been fully tested in conjunction with the Cavitation module.


Cavitation Module Implementation-Model Setup and Solution-Model Options-Introduction

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. All of the model options for the Cavitation Module are located under the Cavitation (Cav) tab. The Model Options section includes:


Model Options-Cavitation (Cav)

Model Options-Liquid

Model Options-on-Condensable Gas Concentration

231


Model Options-Phase Change Coefficients

Cavitation Module Implementation-Model Setup and Solution-Model Options-Shared

There are no settings under the Shared tab affect the Cavitation Module. (See Control Panel-Model Options for details.)

Cavitation Module Implementation-Model Setup and Solution-Model Options-Cavitation (Cav)

All of the model options for the Cavitation Module are located under the Cavitation (Cav)).

Model Options Panel in Cavitation Module Mode

Cavitation Module Implementation-Model Setup and Solution-Model Options-Liquid

In the Liquid region, you are required to provide the operating temperature (K) for the simulation, and the surface tension (N/m) for the liquid. The value that is supplied for operating temperature will be used for all boundary conditions and initial conditions. The default surface tension value is 0.0717 N/m which is the value for water at 300 K.

Cavitation Module Implementation-Model Setup and Solution-Model Options-Non-Condensable Gas Concentration

You may pick the non-condensable gas present in the working fluid. The choices are Air, Helium, Nitrogen, and User Specify. By choosing anything other than user specify, CFD-ACE+ will lookup the molecular weight of the gas. If your non-condensable gas is not listed then you may select User Specify and enter a name for the gas as well as its molecular weight. The molecular weight specified here will be used as shown in equation 7-8.

232

Cavitation Module

The mass fraction of non-condensable gas present in the working fluid must also be specified. The default value is 1.5e-5, which is typical for laboratory water.

It should be noted that the presence of non-condensable gases in liquids is a reality. Even a small amount, e.g., 15 ppm has significant effect on both the physical realism and the convergence characteristics of the solution. The temptations of prescribing zero mass fraction of non-condensable gas should be avoided. For many practical problems, e.g., aerated fluids, equipment with air leakage (suction), etc., higher mass fractions of air may lead to more realistic (accurate) results.

Cavitation Module Implementation-Model Setup and Solution-Model Options-Phase Change Coefficients

The phase change rate coefficients (Ce and Cc) can be specified here. These coefficients are used as described in equation 7-4 and equation 7-5. The default values are Ce = 0.02, and Cc = 0.01. These values have been determined after considerable numerical experimentation over a wide range of flow conditions, for orifice and hydrofoil flows. These values should not be changed without consulting ESI CFD Technical Support. The only exception is that Ce and Cc may both be set to 0.0 to remove phase change effects from the cavitation model if so desired.

Cavitation Module Implementation-Model Setup and Solution-Volume Conditions

Click the Volume Conditions [VC] tab to see the Volume Conditions Panel. See Control Panel-Volume Conditions for details. Before any property value can be assigned, one or more volume condition entities must be made active by picking valid entities from either the Viewer Window or the VC Explorer.

With the volume condition setting mode set to Properties select any fluid volume conditions and ensure that the volume condition type is set to Fluid. Only volume conditions that are of type Fluid need to have Cavitation Module properties specified (since there is no flow in solid or blocked regions there are no Cavitation Module properties for those regions.)

Because activation of the Cavitation Module is currently a global operation, all Fluid volume condition regions in the simulation should have the same volume condition settings.

When performing simulations with the Cavitation Module, the fluid density evaluation method must be set to Cavitation Model for all of the fluid volume conditions. Once the density evaluation method has been set to Cavitation Model an input panel appears.

Volume Condition Inputs for Cavitation Module

233


You are required to provide fluid properties (absolute saturation pressure, liquid phase density, and vapor phase density) at the current operating temperature. The default values correspond to the properties of water at 300 K. These properties should be evaluated at the operating temperature which was specified in the Cavitation Model Options area (see Model Options Settings).

Cavitation Module Implementation-Model Setup and Solution-Boundary Conditions

There are no boundary condition parameters required for the Cavitation Module. The Cavitation Module is fully supported by the Cyclic, Thin Wall, and Arbitrary Interface boundary conditions. See Cyclic Boundary Conditions, Thin-Wall Boundary Conditions, or Arbitrary Interface Boundary Conditions for details on these types of boundary conditions and instructions for how to implement them.

Most simulations will use fixed total pressure inlets and fixed static pressure outlets (see Boundary Conditions). If solution start-up problems are encountered you may want to try starting with a fixed velocity inlet to give sensible limits to the velocities, pressure, density, and turbulence quantities and later switching to a fixed total pressure inlet. See Variable Limits.

In many applications, the cavitation region extends up to the outlet. The common practice of prescribing uniform exit pressure may result into some numerical effects, e.g., pseudo shocks near exit, and some inaccuracy in the computed mass flow rate. In spite of this inaccuracy, you can still study the relative effects of other engineering (geometry and operating flow conditions) parameters. However, to improve the accuracy, it is recommended to extend the calculation domain to locate the outlet boundary condition further downstream such that there is no cavitation region crossing the outlet.

Cavitation Module Implementation-Model Setup and Solution-Initial Conditions

There are no special initial condition settings needed for the Cavitation Module. The vapor fraction will be initialized as zero everywhere.

In difficult cavitation cases it may be beneficial to obtain a nearly converged solution with an increased level of non-condensable gas present (say a mass fraction of 5.0e-5) and then restart from that solution with the desired non-condensable gas mass fraction. See Control Panel-Initial Conditions for details on how to restart a simulation. You may also try to set Cr, Ce, and the non-condensable gas level to zero to obtain a realistic pressure field, and then restart from the solution with the default Ce and Cv, and the desired non-condensable gas level.


Cavitation Module Implementation-Model Setup and Solution-Solver Control-Introduction

Click the Solver Control [SC] tab to see the Solver Control Panel. The Solver Control panel provides access to the settings that control the numerical aspects of the CFD-ACE-Solver and the output options. The Solver Control section includes:




Solver Control-Variables Limits

Cavitation Module Implementation-Model Setup and Solution-Solver Control-Spatial Differencing Scheme

234

Cavitation Module

Under the Spatial Differencing tab, select the differencing method to be used for the convective terms in the equations. Activating the Cavitation Module enables you to set the cavitation vapor fraction calculation. The default method is first order Upwind. See Spatial Differencing Scheme for more information on the different differencing schemes available. Also see Discretization for numerical details of the differencing schemes.

Cavitation Module Implementation-Model Setup and Solution-Solver Control-Solver Selection

Under the Solvers tab you may select the linear equation solver to be used for each set of equations. Activating the Cavitation Module enables you to set the cavitation vapor fraction equation. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 500 sweeps and a convergence criteria of 0.0001. Since the mass vapor fraction typically in the range of 0 - 10-5, it may be beneficial to set the value of the convergence criteria to a much smaller number, perhaps 10-10 or 10-14. See Control Panel-Solver Selection for more information on the different linear equation solvers available. See Numerical Methods-Linear Equation Solvers for numerical details of the linear equation solvers.

Cavitation Module Implementation-Model Setup and Solution-Solver Control-Under Relaxation Parameters

Under the Relaxation tab, select the amount of under-relaxation to be applied for the dependent (solved) variable used for the cavitation vapor fraction equation. See Under Relaxation Parameters for details on the mechanics of setting the under relaxation values. See Under Relaxation for numerical details of how under-relaxation is applied.

The cavitation vapor fraction equation uses an inertial under relaxation scheme and the default value is 0.8. Increasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.


Cavitation Module Implementation-Model Setup and Solution-Solver Control-Variable Limits

Settings for minimum and maximum allowed variable values can be found under the Limits tab. CFD-ACE+ ensures that the value of any given variable will always remain within these limits by clamping the value. Activating the Cavitation Module enables you to set the cavitation vapor fraction. See Variable Limits for details on how limits are applied.

The default min/max for the cavitation vapor fraction is 0 and 1 respectively. These limits should never need to be changed.

Cavitation Module Implementation-Model Setup and Solution-Output

There are no settings under the Output tab that affect the Cavitation Module. See Control Panel-Output Options for details.

Printed Output

There are no settings under the Print tab that effect the Cavitation Module. See Control Panel-Printed Output for details on the general printed output options including boundary condition integral output, diagnostics and monitor point output.

235


Graphical Output

Under the Graphics tab, you may select the variables to output to the graphics file (modelname.DTF). These variables will then be available for visualization and analysis in CFD-VIEW. Activating the Cavitation Module allows output of the variables listed:

Cavitation Module Graphical Output

Variable Units

Total Void Fraction ( = v g)

-

Cavitation Module Implementation-Post Processing

CFD-VIEW can post-process the solutions. When you activate the Cavitation Module, the pressure and void fraction fields can be visualized with surface contours and analyzed through the use of point and line probes. Viewing the void fraction is the most direct indication of the size and shape of cavitating regions in the flow field. The computed mass flow rate and surface pressure distributions are useful for quantitatively assessing performance. A list of Cavitation Module post processing variables is shown below.

Post-Processing Variables


MassFr Vapor Mass fraction -

Total_Volume_Fraction Total volume fraction (Void Fraction)

-

Vapor_Volume_Fraction Vapor volume fraction -

Cavitation Module Frequently Asked Questions

How do I invoke cavitation without phase change (i.e. to simulate effects of mixed non-condensable gas only)?

By setting the phase change rate coefficients (Ce and Cc) to 0.0 (see Phase Change Coefficients) you will not allow phase change (see equation 7-4 and equation 7-5). When this is done the fluid volume condition density evaluation method should still be set to Cavitation Model and the inputs for saturation pressure and vapor phase density will be ignored.

Cavitation Module References

236

Cavitation Module

237

Athavale, M.M., Li, H.Y., Singhal, A.K., “Application of the Full Cavitation Model to Pumps and Inducers, 8th International Symposium on Transport Phenomena and Dynamics of Rotation Machinery.” (ISROMAC-8), Honolulu, HI, March 2000.

Bordelon, Jr., W.J., Gaddis, S.W., and Nesman, T.E., “Cavitation Environment of the Alternate High Pressure Oxygen Turbopump Inducer.” ASME Fluids Engineering Conference, Hilton Head, SC, 1995.

Brennen, C.E., “Cavitation and Bubble Dynamics." Oxford University Press, 1995.

Hinze, J.O., “Turbulence.” McGraw Hill, 2nd Edition, 1975.

Keller, A.P. and Rott, H.K., ”The Effect of Flow Turbulence on Cavitation Inception. ASME FED Summer Meeting, Vancouver, Canada, 1997.

Reisman, G., Duttweiler, and Brennen, C., “Effect of Air Injection on the Cloud Cavitation of a Hydrofoil.” ASME FED Summer Meeting, Vancouver, Canada, 1997.

Rood, E.P., “Critical Pressure Scaling of Schiebe Headform Traveling Bubble Cavitation Inception." ASME FED Summer Meeting, Vancouver, Canada, 1997.

Singhal, A.K., Li, H.Y., Athavale, M.M., and Jiang, Y., “Mathematical Basis and Validation of the Full Cavitation Model.” Proceedings of ASME FEDSM, 2001.

Stoffel, B., and Schuller, W., “Investigations Concerning the Influence of Pressure Distribution and Cavity Length on Hydrodynamic Cavitation Intensity.” ASME Fluid Engineering Conference, Hilton Head, SC, 1995.

Watanabe, M. and Prosperetti, A., “The Effect of Gas Diffusion on the Nuclei Population Downstream of a Cavitation Zone." ASME FED Vol 190, Cavitation and Gas Liquid Flow in Fluid Machinery and Devices, 1994.

Grid Deformation Module

Grid Deformation Module Introduction

The Grid Deformation Module is used by the CFD-ACE-Solver to allow for moving/deforming grid problems. This module is often coupled with the Stress Module to perform full fluid structures interaction problems. The Grid Deformation Module can also impose a known grid deformation for time-dependent moving grid problems. The Grid Deformation Module includes:

Grid Deformation-Applications

Grid Deformation-Features

Grid Deformation-Limitations

Grid Deformation-Implementation

Grid Deformation-Frequently Asked Questions

Grid Deformation-Applications

Grid Deformation Module Applications-Introduction

The Grid Deformation Module simulates fluid flow (gas or liquid) problems where some or all of the boundaries may be in motion. The Grid Deformation Applications section includes:

Applications-Fluid-Structures Interaction Problems

Applications-Simple Prescribed Motion

Applications-User Defined Motion

Grid Deformation Module Applications-Fluid-Structures Interaction Problems

One of the most common uses for the Grid Deformation module is the coupling of the Flow and Stress Modules to perform a fluid structure interaction simulation. In this type of problem, the Grid Deformation Module controls the grid deformation in the fluid regions of the simulation. The Stress Module actually controls the deformation in the solid regions.

Grid Deformation Module Applications-Simple Prescribed Motion

The Grid Deformation Module can perform relatively simple deformation problems. If the boundary motion consists of translation or rotation that can be described by a mathematical expression, then the inputs in CFD-ACE+ will allow the problem to be setup and run directly from CFD-ACE+. For more complex motions, use the user subroutine udeform_bc to define motion for boundaries.

Grid Deformation Module Applications-User Defined Motion

For more complex grid deformation problems, use the UGRID user subroutine to gain total control of the grid deformation and perform very complex deformation problems.

238


Grid Deformation-Features

Grid Deformation Module Features-Introduction

The Grid Deformation Module controls the grid deformation in one or both of two ways: automatic remeshing and user defined remeshing. The Grid Deformation Features section includes:

Features-Automatic Remeshing

Features-User Defined Remeshing

Grid Deformation Module Features-Automatic Remeshing

Automatic Remeshing means that the Grid Deformation Module will automatically remesh the interiors of any structured grid volume conditions whose boundaries are moving. This only applies to structured grid regions of the model. The automatic remeshing feature uses a standard transfinite interpolation (TFI) scheme to determine the interior node distribution based on the motion of the boundary nodes.


Features-Automatic Re-meshing

Transfinite Interpolation Scheme This method uses a standard transfinite interpolation (TFI) scheme to determine the interior node distribution based on the motion of the boundary nodes. This scheme is only available for structured zones. It cannot be applied to composited domains. This must be addressed when building the grid; each zone should only contain one volume condition. It cannot be applied to domains containing cyclic boundary conditions.


Features-Automatic Re-meshing

Solid-body Elasticity Analogy

This method is based on the solution of the equations of linear elasticity. The re-meshing problem is posed as follows: given the set of displacements on the boundaries of a domain, calculate the resulting displacements (and thus mesh movements) of the interior nodes.

The linear elastic equations are derived from a force balance between internal stresses and external forces. These equations may be expressed in terms of displacement as [1]:

(8-1)

Where ui is the ith component of displacement, fi is the ith component of the body force, and and are the Lame constants, expressed in terms of material properties as:

(8-2)

239


(8-3)

In equations 8-2 and 8-3, is the modulus of elasticity and is Poisson’s ratio. For this application, is set to zero to simplify the equations and reduce the cross-equation coupling. Also, the body force is zero since all the displacement results from the specified boundary node displacements. This results in the following equation governing the displacements of the interior nodes.

(8-4)

This equation is solved using finite element formulation. The simple nature of the equation results in a much faster assemble time than standard structural mechanics solvers. Also, with displacement fixed on all or most of the boundaries, the solution is tied down very well and can be solved very quickly using an alternative solver such as the conjugate gradient solver.

Finite Element Solution

Equation 8-4 is solved using a standard Galerkin formulation [2], which can be expressed as

(8-5)

Where is the shape function for node and V is the volume. Integrating by parts and collecting terms gives the following equation:

(8-6)

For the re-meshing problem, the displacement will be specified on the surface and thus the area integral in the above equation, which represents a traction boundary condition, can safely be removed. The internal displacements are interpolated from the nodal displacements using the shape functions:

(8-7)

Where is the jth component of displacement at node . This results in the following equation for nodal displacements:

(8-8)

where

(8-9)

and is the Kronecker delta.

The value of E need not be constant over the entire domain, and in fact can be used to provide extra stiffness in regions of the domain where it is needed. Currently, E is a nonlinear function of the element volume.

Following are the advantages over TFI scheme

240


This scheme applies to both structured and unstructured grids(Tetrahedrals, Prisms, and Pyramids).

It can be applied to composited zones.

Following are some of the known limitations of this scheme.

Computationally (memory and time) expensive compared to TFI. It is more economical to use TFI if applicable.

Polyhedral grid cells are not supported.

The scheme cannot be used in parallel processing due to limitations of the stress solver.

Grid Deformation Module Features-User Defined Remeshing

User defined re-meshing enables you to use the user subroutine UGRID to manually control all of the grid deformation for a given zone. Use this method for structured and unstructured regions for the model. For this, motion must be specified for all the nodes inside the user specified zone. You can use your own re-meshing schemes.

Grid Deformation Module Limitations

The Grid Deformation Module can currently only handle automatic remeshing of structured zones.

Composited zones into single zones is not allowed when Grid Deformation is selected. This must be addressed when building geometry: each zone should contain only one volume condition.

The Grid Deformation Module does not work if the corresponding zone contains cyclic boundary conditions.

The Grid Deformation Module cannot handle composite blocks.

For cases that involve multiple moving boundaries and TFI, each zone must be created using 4 sides in 2D or 6 sides in 3D. Figure 1 and Figure 2 below compare the two gridding options, one that will not work with multiple moving boundaries and one that will.

Grid Deformation-Implementation

241


Grid Deformation Module Implementation-Introduction

The Implementation section describes how to setup a model for simulation with the Grid Deformation Module. The Grid Deformation Implementation section includes:


Model Setup and Solution - Describes the Grid Deformation Module related inputs to the CFD-ACE-Solver

Specialized Point Constraint - Describes how to generate an initial grid for simulating mesh deformations related to CFD applications.


Grid Deformation Module Implementation-Grid Generation

The Grid Deformation Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). All grid cell types are supported (quad, tri, hex, tet, prism, poly). For the automatic remeshing method only structured (quad, hex) grid types are supported.

It is important that while building the grid system that you keep in mind how the deformation will affect the grid. As the boundaries move, the interior will be remeshed using a standard TFI algorithm. If the boundaries move too much or the motion is not well described, then grid quality could be degraded.


Grid Deformation Module Implementation-Model Setup and Solution-Introduction

CFD-ACE+ provides the inputs required for the Grid Deformation Module. Model setup and solution requires data for the following panels:

Problem Type

Model Options

Volume Conditions

Boundary Conditions

Grid Deformation Module Implementation-Model Setup and Solution-Problem Type


Select Grid Deformation to activate the Grid Deformation Module. This module can work with any of the other CFD-ACE+ modules.


Grid Deformation Module Implementation-Model Setup and Solution-Model Options-Introduction

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. The Model Options section includes:


242


Model Options-Deform

Model Options-Auto Remesh

Model Options-User Subroutine

Grid Deformation Module Implementation-Model Setup and Solution-Model Options-Shared

There are no settings under the Shared tab that are related to the Grid Deformation Module, although most Grid Deformation problems will be run in the transient mode. (See Model Options for details.)

Grid Deformation Module Implementation-Model Setup and Solution-Model Options-Deform

All of the model options for the Grid Deformation Module are located under the Grid Deformation (Deform) tab.

Model Options Panel in Grid Deformation Module Mode

There are two options available; auto remesh, and user subroutine (ugrid). At least one needs to be chosen and both may be activated together if desired.

Grid Deformation Module Implementation-Model Setup and Solution-Model Options-Auto Remesh

The default option for the Grid Deformation Module is to automatically remesh the structured grid zones based on the motion of the boundaries. This can only be done for structured grid zones.

User Subroutine

If you select the user subroutine option, the CFD-ACE-Solver will call the user supplied subroutine (ugrid) that enables you to control each grid node’s location manually. See User Subroutine for details on using the UGRID subroutine.

Grid Deformation Module Implementation-Model Setup and Solution-Model Options-User Subroutine

If you select the user subroutine option, the CFD-ACE-Solver will call the user supplied subroutine (UGRID) that enables you to control each grid node’s location manually. See User Subroutine-UGRID for details.

Grid Deformation Module Implementation-Volume Conditions

243


Volume condition settings are only needed if the User Subroutine grid deformation option was chosen under the Model Options panel.

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. See Control Panel-Volume Conditions for details. Before any volume condition information can be assigned, one or more volume condition entities must be made active by picking valid entities from either the Viewer Window or the VC Explorer.

For each volume condition that you want to control grid deformation through the UGRID user subroutine, you must activate the Moving Grid flag. This is done by changing the Volume Condition setting mode to General and then selecting the Moving Grid checkbox for each volume condition region that you wish to control via the UGRID subroutine. See User Subroutines-UGRID for details on how to implement the UGRID subroutine.


Grid Deformation Module Implementation-Boundary Conditions-Introduction

Click the Boundary Conditions [BC] tab to see the Boundary Condition Panel. See Control Panel-Boundary Conditions for details. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or the BC Explorer.

All of the general boundary conditions for the Grid Deformation Module are located under the Grid Deformation (Deform) tab and can be reached when the boundary condition setting mode is set to General. Each boundary condition is assigned a type (e.g., Inlet, Outlet, Wall, etc.). See Control Panel-BC Type for details on setting boundary condition types.

Because the Grid Deformation Module deforms the grid, it does not need any boundary condition values, but rather it needs to know how to move the boundary condition locations. For this reason, the boundary condition types do not matter (i.e., the boundary condition description below applies to all types of boundary conditions).

The boundary conditions necessary to simulate the translation or rotation of any boundary condition patch are available by selecting the Grid Deformation (Deform) tab. There are two methods used to define the motion; translation and rotation. By combining the translation and rotation methods, different moving patterns can be modeled, such as deformation and wave motion.

The Grid Deformation Boundary Conditions section includes:

Boundary Conditions-Translation

Boundary Conditions-Rotation

Grid Deformation Module Implementation-Boundary Conditions-Rotation

There are two rotation subtypes: Rotation Angle and Angular Velocity. Each option is exclusive of the other for a specific face. However, you can specify both rotation and translation for a face. The table provides a summary of the rotation variables.

Rotation Input Variables

Variable Description

244


point1 x, point1 y, point1 z Coordinates of the first point of rotation axis. These are real values.

point2 x, point2 y, point2 z Coordinates of the second point of rotation axis. These are real values.

Forward Angle

Forward angle in degrees. Measured between the initial and final positions.

Backward Angle

Backward angle in degrees. Measured between the initial and final positions opposite the Forward Angle

.

Rotation Angle Rotation angle in degrees (for Rotation Angle mode only).

Rotation Omega Rotation annular velocity in degree/second (for Angular Velocity mode only)

For 2D geometry, CFD-ACE+ only displays point1 x and point1 y since the rotation of 2D geometry must be perpendicular to the screen, that is in the z-direction.

The Rotation Angle or Rotation Omega inputs are input with constant values or mathematical expressions or UserSub(udeform_bc).

The forward and backward angles are defined as rotation for a rotating plane. By specifying Forward Angle and Backward Angle, a face can oscillate anywhere. They could be same and/or different. If Forward Angle = Backward Angle, a face will stop rotation at Forward Angle.

245


The rotating plane will rotates between a and b. The angle a or b is called forward or backward angle. If a is defined as forward angle, then b will be as backward angle and vice versa.

Grid Deformation Module Implementation-Specialized Point Constraint

Tips on Moving Grid Setup

The capability of moving grids in CFD-ACE+ has provided a great opportunity to simulate mesh deformation-related CFD applications. These kinds of applications exist in solid-fluid interaction, bio-membranes, and MEMS devices. However, it is also known that the setup process is not a trivial job especially for complicated geometries. This technical note provides guidance on how to generate an initial grid so that the software can provide you with expected simulation answers.

Domain Division

Correct domain division of geometry within CFD-GEOM is the first, important step in handling the moving grid cases. CFD-ACE+ uses the linear interpolation algorithm to generate new grids dynamically. This technique requires that you generate an initial grid through CFD-GEOM correctly by domain division. The following figure shows the basic structure of a domain.

Figure 1 - Domain Definition - Points a, b, c, and d define a domain. a, b, c and d are

domain corner points, points w, e, s and n are domain face points. ab, bd, dc and ca are edges. Internal point (i, j) will be moved by the linear interpolation algorithm.

246


The linear interpolation algorithm works in the following order:

(1) From motions of corner points, calculating face point motions, i.e., from motions of corner points a, b, c, and d, getting the motions of face points w, e, s, and n.

(2) From motions of face points, calculating internal point motions, i.e., from motions of face points w, e, s and n, get the motion of internal point (i, j).

From the order above, we can see that the end point of the prescribed moving edge (or face in 3D) must be corners of the domain.

Figure 2 shows a correct domain structure. Since the edge ac is moved by the prescribed motion, points a and c are corner points of domain abcd. The final new grid will look like figure 3.

Figure 2 - Moving Edge ac - Point a and c are corner points of domain.

Figure 3 - Final Grid After Motion

Figure 4 is still a correct domain structure. The edge ab has been divided into two edges: edge ae and edge eb. However, the end points of the edge with prescribed motion ac are points a and c, which are the corner points of the domain abcd. The final new grid will look like figure 5.

247


Figure 4 - Moving Edge ac - Point a and c are corner points of domain. ae and eb are

edges.

Figure 5 - Final Grid After Motion

However, figure 6 defines an incorrect domain structure. Since the end points of moving edge ae are points a and e, e is not the corner point of domain abcd.

Figure 6 - Moving Edge ae. Point e is not a corner points of domain abcd.

To get the correct grid, the domain should be divided as in figure 7. Where point e becomes corner point of domain aee’c. The final grid will look like figure 8.

248


Figure 7 - Moving Edge ae. Point e is a corner points of domain aee’c.

Figure 8 - Final Grid After Dividing Domain

Figure 9 shows the multi domain structure.

Figure 9 - Multi Domain Structures

249


However, the structure shown in figure 9 is not correct! The whole structure has been divided into 4 domains. The prescribed motions are assigned to the edges gj and hi of domain 2. The end points of the moving edge are points h, i, g and j. These points are the corner points of domains 1, 2 and 3 respectively. However, the moving points i and j are also shared by domain 4. They are not the corner points of domain 4! Therefore, the structure in figure 9 will fail to return the correct grids. The correct structure is drawn in figure 10.

Figure 10. Multi Domain Structures

In the figure 10 the moving points i and j are also the corner points of domains 4, 5 and 6 respectively.

Using the .spc File

SPC is the acronym for specialized point constraint and is a CFD-ACE+ file type. Using the structure in figure 10, we will explain the .spc file. We assume that the moving edges gj and hi are moving toward an upper vertical direction. After the motion, the new grid should look like figure 11.

Figure 11 - New Grid After Motion

Since the points t and s do not move, the new grid on the right becomes skewed. Occasionally, the skewed grids create problems in numerical simulations and may lead to unstable schemes or delay the convergence. If the skewed grid becomes an issue, the .spc file can repair the grid. With the .spc file, you can define the motions of points t and s the same as points j and i so that the new grid becomes orthogonal as shown in figure 12.

250


Figure 12 - With the .spc File, the New Grid Becomes Orthogonal

Another feature of the .spc file is that you can define a 1D re-meshing algorithm. The moving grid re-meshing algorithm uses the linear TFI (Transfinite Interpolation) methodology.

Figure 13 - Grid Structure and TFI Methodology

The motion of internal grid point (i, j) is calculated based on the motions defined at edges through the 2D re-meshing:

Where, dx and dy are displacements, function f is a linear interpolation function that is defined by the ratio of the edge length. Sometimes if the geometry has one or more edges that are extremely non-linear, as shown in figure 14, the linear 2D re-meshing method may not work.

251


Figure 14. Highly Non-linear Geometry. The motion is along the vertical direction (Y

direction)

If we still use 2D re-meshing to calculate the motion of internal point (i, j), the grid will create a negative volume. This is because the non-linear edge ratio at points e and w cannot be interpolated by linear function f. We can use the 1D re-meshing technique to repair it. With 1D re-meshing, the motion of point (i, j) is calculated only through displacement in the Y direction, i.e., based on the motion at points n and s:

We can do this using the .spc file. The .spc file has 4 different types of techniques:

1D re-meshing

Sliding

Face to point re-meshing

Point to point re-meshing

The function of "sliding" and "face to point re-meshing" can be achieved by "point to point re-meshing". Therefore, the point to point re-meshing is more general. Its format is:

Zone(n1, n2) : i1, j1, k1, i2, j2, k2, X_dir, Y_dir, Z_dir

Where Zone is the key word. n1 and n2 are the domain index. The i1, j1, an k1 are grid node indexes on domain n1, the i2, j2, and k2 are grid node indexes on domain n2. X_dir, Y_dir and Z_dir are the re-meshing direction. Based on the problem, you may only need one or two of them:

Zone(n1, n2) : i1, j1, k1, i2, j2, k2, Y_dir

Zone(n1, n2) : i1, j1, k1, i2, j2, k2, X_dir, Y_dir

The zone n1 is the follower zone and n2 is the leading zone. The function of this specification defines an assignment, i.e. the displacement of point (i2, j2, k2) on the domain n2 is assigned to the point (i1, j1, k1) on the domain n1:

The detailed format is written in the .spc help file.

252


253

If you open a blank .spc file as model.spc, where model is the DTF file model name, and run the simulation, CFD-ACE+ will stop the calculation and write a .spc help file. Follow the hints in help file to generate a working .spc file and run the simulation again.

Grid Deformation Module Implementation-Post Processing

CFD-VIEW can post-process the solutions. When the Grid Deformation Module is invoked, the deformed grids will be written to separate simulations in the DTF file.

Grid Deformation Module Frequently Asked Questions

What happens if a node shared by two boundaries has two different types of motions?

To illustrate the basic rules that CFD-ACE+ follows, consider the following figure:

If node 1 is shared by being (translation or rotation) and bc3 (implicit Motion), and Stress or Free Surfaces (VOF) are being solved for, then node 1 uses Implicit Motion.

If stress of VOF is not being solved for, then node1 uses Translation or Rotation type.

The Order of Preference is:

1. Implicit Motion (if solving for Stress of Free Surfaces (VOF))

2. Normal Translation Motion

3. UserSub(udeform_bc)

4. Regular Translation or Rotation

If both nodes have the same type of motion but expressions are different, then CFD-ACE+ writes a warning message to <model-name>.out file and one of them is chosen. If expressions are the same, then no warning message is given. Corresponding warning messages for each decision will be written to <modelname>.out file.

Stress Module

Stress Module Introduction

The Stress Module adds a finite element structural analysis capability to CFD-ACE+ and enables you to set up the structural model. You can use it in a stand-alone mode for pure structural analyses or couple it with Flow, Heat Transfer, and Electric modules for multi-disciplinary analyses. These multi-disciplinary analyses may be grouped into two different categories.

Implicit coupling with other modules is accomplished by sending temperatures, fluid pressures, electrostatic pressures, etc. to the Stress Module. The Stress Module calculates deformations (and stresses) from these loads and updates the geometry and grid. Iterations are performed until convergence is obtained. You have control over how often the geometry/grid is updated via the stress modules by specifying a grid update frequency. The Stress Module includes:

Stress Module-Applications

Stress Module-Features

Stress Module-Theory

Stress Module-Limitations

Stress Module-Implementation

Stress Module-Frequently Asked Questions

Stress Module-Examples

Stress Module-References

Stress-Applications

Stress Module Applications Introduction

The implicit coupling provided among the Stress and Flow, Heat Transfer, and Electric modules provides a very powerful and wide ranging analysis tool. The Stress Applications section includes:

Stress Module-Pure Structural Analysis

Stress Module-Coupled Solid/Fluid/Thermal Problems

Stress Module-Multidisciplinary Electrostatic Problems-MEMS Application

Modules-Stress Module Application-Pure Structural Analysis

The Stress Module is a structural analysis tool for calculating stress and deflection. The figure below shows the results of thermoelastic analysis of fuel transfer tubes in a gas turbine atomizer. A thermal analysis was not performed in this problem; rather, a constant temperature increase of 500K was applied to the geometry, and the resulting stresses calculated.

254

Stress Module

Thermoelastic Analysis of a Gas Turbine Atomizer

Stress Module Applications-Coupled Solid/Fluid/Thermal Problems

You can perform problems involving the interaction of flow, heat transfer, and stress analysis using CFD-ACE+, without transferring external data files between different analysis packages.

The figure below shows a static mixer channel used for mixing turbulent fluids. For large flow rates, stresses at the junction of the mixer arms and the base can become quite large. For this problem, the deflection of the mixer arms is not large enough to have an impact on the flow field.

The grid and analysis results for this problem is given in the following diagram. This problem used the one-way coupling feature, in which a converged steady-state flow field was obtained, and then the pressure loading from that flow field was applied to the Stress Module to calculate the resulting stresses.

Mixer Channel Geometry

255


Mixer Channel Grid and Solution

The following image shows the results of a coupled fluid/thermal/structural analysis of flow through an orifice. The geometry is modeled as 2D axisymmetric. Hot gas enters the left side of the domain at 0.25 m/s and 500K. The initial temperature of the orifice was 300K. This analysis used the two-way coupling option, where the geometry deformations from the structural analysis were fed back to the Flow and Heat Transfer Modules.

It also shows the results of four different analyses run for this problem. Image A shows the results with no structural analysis, i.e. running just a flow and heat transfer problem. Image B and Image C show the results of coupling the Stress Module with the Heat Transfer and Flow Modules only, respectively. Image D shows the results of the fully coupled problem (flow, heat transfer, and stress). This problem shows the necessity of performing this fully coupled solution, since the results of the flow plus heat transfer solution is not a simple linear combination of the individual flow and heat transfer solutions (because of the nonlinearity introduced by the flow solution).

Coupled Fluid/Thermal/Structural Analysis of an Orifice

Stress Module Applications-Multi-disciplinary Electrostatic Problems-MEMS

You can analyze MEMs applications by coupling the CFD-ACE+ Stress and Electric Modules with the Flow and Heat Transfer Modules. Several examples are shown here. The figure below

256

Stress Module

depicts a model of an accelerator, which is an electrostatic loaded plate clamped by four beams. This plate sits 2m above a ground plane, and has a 20V voltage applied to it. This figure shows the calculated displacement contours resulting from the electrostatic load.

Accelerometer Under an Electrostatic Load

Image A shows the geometric dimensions and problem set-up. Image B shows the calculated displacement of the plate due to the electrostatic load. The displacement of the plate toward the ground plane is maximum (1.83 mm) at the center of the upper plate.

The following figure shows a high frequency resonator, used in applications such as high pass filters. A sinusoidal driving voltage is applied to a plate below a resonator beam, deforming it as seen by the contours. The deformation is coupled through a coupling beam to an output beam where the change in capacitance between the beam and the ground detects its deformation. The contours show the calculated vertical displacement for one instance in time.

Displacement Field Contours on a High Frequency Resonator

The following figure shows a linear lateral resonator comb drive. The device has the potential for many uses such as an accelerometer or gyroscope. A folded beam with attached combs is

257


placed between comb drives which have applied ac or dc voltages to drive or sense the resonance of the moving folded beam structure.

The following figure shows the structure at two instances in time. The plotted contours represent the vertical displacement (i.e. normal to the ground plane). The contours show the resultant increase in tilt due to the electric field asymmetry in the vertical direction as the voltage is increased.

Linear Lateral Resonator Comb Drive with an Applied Sinusoidal Drive Voltage at Two

Instances in Time

The following figure shows a mesopump where pump actuation is accomplished through electrostatic forces. This analysis was a flow/stress/electrostatic analysis. The figure that follows shows the computed electric field distribution.

258

Stress Module

Solid Model of a Mesopump Cell

Electrostatic Field Distribution in a Pump

Stress Module Features

The Stress Module supports the following structural analysis capabilities:

Steady-state and transient analysis

Linear analysis

259


Nonlinear analysis: geometric nonlinear, material nonlinear

Contact analysis: elastic/rigid and elastic/elastic

Piezoelectric (for details see Electric Module)

Thermoelasticity

Modal analysis

Anisotropic material properties

Various element types

Stress-Theory

Stress Module Theory-Introduction

The Stress Module solves the structural mechanics equations, in finite element form, derived from the principal of virtual work (Zienkiewicz, 1971). For each element, displacements are defined at the nodes and obtained within the element by interpolation from the nodal values using shape functions. In matrix notation, this may be expressed as:

(9-1)

where u is the continuous displacement field throughout the element, N is the shape function matrix, and a is the vector of nodal displacements. The particular form of N depends on the element type and order. Using the strain-displacement relationship, the strains are derived from the displacements u and hence the nodal displacements a. This may be expressed as:

(9-2)

If the displacements are large, the strains depend non-linearly on the displacements and thus B is a function of a. We express this relationship as:

(9-3)

Where B0 is the standard small-strain strain-displacement matrix, and BL is a linear function of the nodal displacement.

For reasonably small strains, the stress-strain relationship is linear and may be expressed as:

(9-4)

Here, D is the elasticity matrix containing the material properties, 0 and 0 are initial strains and stresses, respectively.

Thermoelastic stress problems are handled by considering the temperature rise T to contribute to initial strains as:

260

Stress Module

(9-5)

where i represents the coefficient of thermal expansion for coordinate direction xi.

The governing equations are derived by forming a balance between the external and internal generalized forces using the principal of virtual work. If we let f be the vector of externally applied loads, and apply a nodal virtual displacement of a, the work done by the external and internal forces, respectively, are:

(9-6)

(9-7)

where equation 9-2 was used in expressing the strain in terms of the nodal displacements. Equating the external work done with the total internal work, and recognizing this equality must be valid for any value of virtual displacement, we arrive at the following equilibrium equation.

(9-8)

For the nonlinear case, a Newton-Raphson technique is used where at each iteration we solve for a correction to the current displacement field using:

(9-9)

The rate of change of with respect to a is defined as the tangent stiffness matrix, KT. Taking variations of equation 9-8 with respect to da gives:

(9-10)

From the stress-strain relationship (equation 9-4) and equation 9-2, we can write:

(9-11)

and from equation 9-3:

(9-12)

Thus:

261


(9-13)

where:

(9-14)

The integral in equation 9-13 may be written as (Zienkiewicz, 1971):

(9-15)

where K is known as the initial stress matrix or geometric matrix.

Thus, the equation solved in the Newton-Raphson scheme is:

(9-16)

where:

(9-17)

Convergence is obtained when the maximum correction a reaches a predetermined small value. For a linear problem, KT is the standard linear stiffness matrix and only one iteration is needed. The number of iterations needed in the nonlinear case is highly problem dependent; typical values range from 3 to 20. For transient analyses, the equilibrium equation (equation 9-8) is modified to account for the inertial and damping forces, and the same procedure is followed to derive the basic equation of the iterative scheme.

Although the general elasticity relationship given by equation 9-4 was used, this approach is general to allow for any nonlinear stress-strain relationship, since the solution will again reduce to the solution of a set of nonlinear equations as expressed in equation 9-8.

Stress Module Theory-Damping

To model structural damping, CFD-ACE+ uses a spectral damping method whereby viscous damping is incorporated by specifying a percent (or fraction) of critical damping (see equation 9-18 below). Critical damping is defined as the transition between oscillatory and non-oscillatory response (see following figure).

262

Stress Module

Stress - Damping Responses

The damping fraction depends on the material and the stress level. These values can be obtained by experimental observations of the vibratory response of a structure, or from past experience with similar structures. Typical values fall between 0.5% and 15% (see equation 9-19).

CFD-ACE+ uses a specific spectral damping scheme known as Rayleigh or proportional damping. This approach forms the damping matrix C as a linear combination of the mass and stiffness matrices. (see equation 9-20)

(9-18)

where and are the mass and stiffness proportional damping coefficients respectively. With this formulation, the critical damping fraction, as a function of frequency, may be express as:

(9-19)

The two damping coefficients and are obtained by specifying fractions of critical damping ( 1 and 2 ) at two frequencies (1 and 2). This yields two equations in two unknowns which may be solved as:

263


(9-20)

(9-21)

If the values of and are known, they may be entered directly in CFD-ACE+. Otherwise the four values, 1, 1, 2,and 2 may be specified, and and will be calculated internally.

The frequency values 1 and 2 are usually chosen to bound the design spectrum of the problem. In such a case, 1 is taken as the lowest natural frequency of the structure (which may be obtained from a modal analysis) and 2 is taken as the maximum frequency of interest in the loading or the response.

As can be seen from equation 9-21, damping attributed to M decreases with increasing frequency, whereas the K component increases with increasing frequency.

The following figure, taken from Cook, Malkus, et al, shows the fraction of critical damping as a function of frequency. The frequency range of interest for this example ranges between 1 and 2 with critical damping fractions 1 and 2 respectively. This figure demonstrates why 1 and 2 are chosen to bound the design spectrum, as the amount of damping increases substantially outside this range.

264

Stress Module

Fraction of Critical Damping versus Frequency for Raleigh Damping. Contribution of Stiffness and Mass Proportional Damping to Total Damping Included

Stress-Limitations

Stress Module Limitations-Limited Element Library

The elements supported by FEMSTRESS are: triangles, quadrilaterals, tetrahedrals, hexahedrals, prisms, pyramids, shells, solid shells, and enhanced bricks. The Stress Module does not support other specialized elements such as beams, bars, rods, and shear panels.

Stress Module Limitations-Arbitrary Interfaces

The grid resolution required in the structural domains is substantially less than that required for the fluid domains. This is especially true when second order elements are used in the solid. However, nodal points must be matched one-to-one at a solid/fluid interface. To work around this, use unstructured elements in the solid to transition from the fine interface grid to a coarser grid away from the interface.

Stress Module Limitations-Cyclic/Thin Wall Boundary Conditions

Stress calculations cannot be made for a thin wall boundary condition.

Stress-Implementation

Stress Module Implementation Introduction

The Implementation section describes how to set up a model for simulation using the Stress Module. It includes the following sections:


Model Setup and Solution - Describes the Stress Module related inputs to CFD-ACE+

Post-Processing - Provides tips on what to look for in the solution output

Stress Module Implementation-Grid Generation

The Stress Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). CFD-ACE+ supports quad, tri, hex, tet, pyramid, and prism cell types, but polyhedral cells are not allowed. CFD-ACE+ can use many different types of elements when solving stress simulations using the built-in finite element stress module.

Element Types

By default, the elements created in a standard grid system, e.g. triangles, hexahedrals (bricks), etc., are taken as first order elements, which means that the dependent variables are interpolated linearly between the nodes.

Alternatively, you may choose second order elements. The stress solver will create second order elements by inserting mid nodes along each edge. Thus, a 3-noded triangle becomes a 6-noded triangle; an 8-noded hexahedral (brick) becomes a 20-noded hexahedral (brick), etc. With second order elements, variables are interpolated quadratically using the three nodes along an edge,

265


greatly increasing the accuracy in most cases. Second order elements should be used with caution, however, because in addition to increasing the accuracy, the memory and computational requirements are also increased.

To improve the accuracy of first order elements, we have added two element types in V2003. You can now activate enhanced first order brick elements or solid shell brick elements. Enhanced first order brick elements are almost as accurate as second order elements with the advantage that they require roughly the same memory as first order elements. Solid shell brick elements are more accurate than standard elements for high aspect ratio grids. CFD-ACE+ can also stack solid shell elements in multiple layers but these options can only be applied to 3D hexahedral elements.

The following example illustrates the different solutions that can be obtained using the different element types. This case is a simple cantilever beam with an applied pressure of 200000 Pa to the upper side of the beam. The applied pressure causes the beam to deflect downward. The analytical result for this case indicate a maximum displacement of 0.0146 m. The results are shown below along with normalized CPU time and memory usage for each element type.

266

Stress Module

It is clear from the results that the first order elements are not accurate enough to capture the correct solution. The second order elements produce the analytical result but take 4x more CPU time and 7x more memory. The enhanced first order elements take roughly the same amount of CPU time and memory for this small case and produce an excellent result.

Choosing An Element Type

First order elements are the most robust and efficient and can be used for many simulations. However, here are a few guidelines that might help:

If you have a bending dominated problem (like the bending of a plate or beam) then second order elements (or enhanced first order bricks) should be used.

If (nearly) incompressible behavior is present (e.g., in linear elastic materials with Poisson's ratio greater than 0.49 or nonlinear elastoplastic materials) then enhanced first order elements or even solid shell elements will perform better than standard first or second order elements. Keep in mind that these element types are only available for 3D hexahedral grid systems.

If you have high aspect ratio hexahedral grids (as in modeling a large plate structure) then you should use solid shell brick elements.

To set an element type:

1. Select Model Options (MO), and then the Stress Tab.

2. In the Element Order pull-down menu, select one of the following options:

First Order

Second Order

To set enhanced first order brick elements or solid shell brick elements:

1. Select Model Options(MO), and then the Stress tab.

2. Set the Element Order to First (this is the default setting).

267


3. Activate the Element Conversion option (this is not required for Enhanced Brick option) if you want Solid Shells.

4. Select a volume condition region that consists of hexahedral cells.

5. Change the VC setting mode to Stress.

6. Change the Brick Element Option to either Enhanced Brick or Solid Shell. If you are using Solid Shell elements then you also need to set the shell surface direction by selecting one BC for each solid that represents the side which will undergo the most bending (usually the surface with the largest area) and activating the Shell Surface option. Enhanced and solid shell options can only be applied to 3D hexahedral grid systems.

Structural Analysis

To perform a structural analysis on a region of the domain:

1. Specify the cells within that region as solid cells by assigning a solid type Volume Condition (VC) to the region.

2. Activate stress in the same region.

If you have activated the Deformation Module, there is another important aspect of grid generation to be aware of. The deformation module uses a transfinite interpolation (TFI) procedure to re-grid the deformed fluid regions. This TFI algorithm requires a structured grid, so any fluid regions of the domain that will be deforming must be structured (i.e. quadrilaterals or hexahedrals).

The following diagram shows the geometry and grid for flow over a solid block. The fluid grid in the immediate vicinity of the solid must be structured if the grid is deforming, but the solid grid itself and the fluids grid away from this region may be unstructured. The interface between the structured and unstructured regions in the fluid is taken as fixed when the TFI is performed.

268

Stress Module

Sample Grid for Analysis of Deforming Grids


Stress Module Implementation-Model Setup and Solution-Introduction

The Model Setup and Solution section includes the inputs and settings required for the Stress Module. It includes:

Problem Type

Model Options

Volume Conditions

Boundary Conditions

Initial Conditions

Solver Control

Output

Stress Module Implementation-Model Solution and Setup-Problem Type

269



Select Stress to activate the Stress Module. The Stress Module can work with any of the other CFD-ACE+ modules. If the grid is deforming (i.e., two-way coupling), then you must also activate the Grid Deformation Module.


Stress Module Implementation-Model Setup and Solution-Model Options-Introduction

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. The Stress Model Options section includes:


Model Options-Stress

Stress Module Implementation-Model Setup and Solution-Model Options-Shared-Introduction

The Model Options Shared tab contains parameters that apply to all modules. For FEMSTRESS, all but the Time Accuracy option apply, since FEMSTRESS always uses the Newmark algorithm for time advancement. See Control Panel-Model Options for details in setting up parameters.

Model Options-Stress

Stress Module Implementation-Model Setup and Solution-Model Options-Stress-Introduction

All of the model options for the Stress Module are located under the Stress tab. The Stress tab includes:

Stress-Analysis

Stress-2D Geometry

Stress-Element Order

Stress-Mass Matrix Type

Stress-Damping

270

Stress Module

Model Options Panel - Stress Tab

Stress Module Implementation-Model Setup and Solution-Model Options-Stress-Analysis

The Stress Tab Analysis section contains a pull-down menu with the following options:

Linear Analysis - In this option, the Modal Analysis option becomes available at the bottom of the panel and enables you to specify the number of modes. If you enter a value N, the Stress Module will calculate the first N mode shapes and their respective frequencies. If you specify the problem as steady, the mode shapes and frequencies will be calculated and written to the output file. If you specify the problem as transient, these values will be used for the transient solution. You can select graphical output of the mode shapes using the Solutions/Output options. Modal analysis can be used with any grid type supported by the stress module. However, this option may only be used for linear problems.

Non-Linear Analysis - In this option, additional non-linear options become available:

o Geometric Non-Linear - The Geometric Non-Linear options activates Large Deformation Analysis which allows the non-linear component of the strain tensor to be considered during formation of the stiffness matrix.

271


o Contact Analysis - If there will be contact between an elastic body and a rigid surface or between two elastic bodies, then activate the Contact Analysis button. Contact problems are nonlinear. Contact and target surfaces are specified in Boundary Conditions.

o Material Non-Linear - This option enables you to solve problems where the modulus of elasticity is a function of strain, as opposed to a default constant value.

Stress Module Implementation-Model Setup and Solution-Model Options-Stress-2D Geometry

For two-dimensional problems, the underlying modeling assumption for structural analysis is either axisymmetric, plane stress, or plane strain. You can choose the axisymmetric geometry option under the Model Options-Shared tab.

If the problem is not axisymmetric, choose:

Plane Stress -If one dimension is smaller than the other dimensions, then the stress in that direction is negligible. This is referred to as plane stress.

Plane Strain - If one dimension is larger than the other two, then the strain in that direction is negligible. This is referred to as plane strain and the analysis is done on a sliced plane of unit thickness in the larger dimension.

Stress Module Implementation-Model Setup and Solution-Model Options-Stress-Element Order

The Element Order menu contains two options:

First - The elements created in a standard CFD grid system, e.g. triangles, hexahedrals, etc., are first order structural elements. This means that the dependent variables are interpolated linearly between the nodes.

Second - The Stress Module will create second order elements out of the first order ones by inserting mid nodes for each edge. Thus, a 3-noded triangle becomes a 6-noded triangle, an 8-noded hexahedral becomes a 20-noded hexahedral, etc. This is only done for the structural analysis. With second order elements, variables are interpolated quadratically using the three nodes along an edge, greatly increasing the accuracy in some cases. Second order elements should be used with caution because in addition to increasing the accuracy, it also increases memory requirements.

To learn more about element types, see Stress-Implementation-Grid Generation.

Stress Module Implementation-Model Setup and Solution-Model Options-Stress-Mass Matrix Type

The Mass Matrix Type menu enables you to setup modal and/or transient problems and contains two options:

Consistent - In the consistent option, the mass matrix is calculated using the standard mathematical finite-element formulation.

Lumped - In the lumped option, the mass matrix is converted to a diagonal matrix by lumping the element mass at the nodes.

For most problems, these two options give nearly identical results.

Stress Module Implementation-Model Setup and Solution-Model Options-Damping

The Damping section enables you to select either structural or numerical damping.

272

Stress Module

Structural Damping

For structural analyses, structural damping may be specified by selecting one of the following methods:

Raleigh Damping - In the Rayleigh Damping method (Bathe, 1976), the damping matrix is a linear combination of the stiffness and mass matrix. The weighting factors are specified in the Alpha and Beta type-in fields. The damping matrix will be computed as:

(9-22)

where:

k = the stiffness matrix

m = the mass matrix

If the values of and are known, you can enter them directly under the Rayleigh Damping parameters.

Critical Damping - In Critical Damping, these parameters define a percent of critical damping at two different frequencies. The values of and are then calculated from this information, as explained in the Stress-Theory section.

Numerical Damping

The transient finite element equations are solved using the Newmark Scheme (Cook, 1981). This method uses a parameter, , that controls the implicitness of the algorithm. For values of , artificial positive damping is introduced—a feature that is used in the Implicit Damping approach.

Allowable values are 0.5< < 1.0. At = 0.5, no artificial damping is added to the solution.

Model Setup and Solution-Volume Conditions

Stress Module Implementation-Model Setup and Solution-Volume Conditions-Introduction

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. To assign volume conditions and activate additional panel options, select an entity from the viewer window or the VC Explorer.

The Stress Module requires two types of volume condition information to be entered: stress equations for a given volume condition and structural properties for a solid material. The Volume Condition mode determines which piece of information is currently being set. The Volume Condition section includes:

Volume Conditions-Activating Stress Equations

Volume Conditions-Structural Properties

Stress Module Implementation-Model Setup and Solution-Volume Conditions-Activating Stress Equations

To activate stress equations solution:

1. In VC Setting Mode, select the Stress option from the pull-down menu.

273


2. If the volume condition type is Solid, you will be able to select the Activate Stress checkbox. If the Activate Stress checkbox is not selected, the solid volume will not be part of the Femstress Solution. In other words, it will be a solid volume with no displacement and no stress. When you activate the Activate Stress checkbox, an additional panel appears:

Volume Condition Panel in Stress Mode

3. In the Tref field, enter a reference temperature for the volume condition that will be used in the thermoelastic analysis. The solver assumes that the given grid is stress free at this reference temperature. If you do not want to include the effects of thermal stress, then set the thermal coefficient of expansion to zero. See Volume Conditions-Structural Properties.

The Heat Transfer Module does not need to be activated for a thermoelastic analysis. For cases where the Flow or Heat Transfer Module has not been activated, the solid temperature will default to 0K. Thus, to model a temperature increase of 100K, the reference temperature must be specified as -100K.

Stress Module Implementation-Model Setup and Solution-Volume Conditions-Structural Properties

With the Volume Condition tab set to Properties, select the volume conditions with the volume condition type set to Solid. Only volume conditions that are a Solid type need to have structural properties specified (since there is no stress in fluid or blocked regions there are no structural properties for those regions.) Structural properties are located under the Struct tab.

Material Type

The Material Type menu enables you to choose whether the material type is linear or non-linear.

Linear Isotropic Materials

The Material section menu enables you to choose an isotropic or anisotropic material. For isotropic materials, where the structural properties are independent of direction, the Stress Module requires you to select four properties:

Density - kg/m3 (Available under the Phys tab)

Young's Modulus (modulus of elasticity) -N/m2

Poisson's Ratio - dimensionless

Expansion Coefficient (coefficient of thermal expansion) - m/m-K.

274

Stress Module

Material Property Inputs for Linear Isotropic Materials

Linear Anisotropic Materials

If you choose the Linear Anisotropic Material type, the panel changes to the following for 3D geometries:

275


Material Property Inputs for Linear Anisotropic Materials

For anisotropic materials, the properties are dependent on direction. For each volume, you must define a local coordinate system in which the properties are defined. This local system is defined by the Local X-Axis Vectors and Local Y-Axis Vectors input. These vectors define the local x and y axes in terms of the global coordinate system. The local z-axis is obtained from the x and y axes using the right hand rule. These vectors do not need to be unit vectors. The number aij in these fields represent the jth component (in the global system) of local axis i. The figure below shows the local system (xy) and global system (XY).

Global (XY) axis and Local Material (xy) Axes

To define this system, enter the following values:

a11 = 1.0 a12 = 1.0 a13 = 0.0 a21 = -1.0 a22 = 1.0 a23 = 0.0

276

Stress Module

After defining the local coordinate system, you must enter values for the expansion coefficient, Young's Modulus, Poisson's Ratio, and Shear Modulus in the local system. Here, index 1 refers to the local x axis, index 2 to the y axis, and index 3 to the z axis.

For a general 3D anisotropic analysis, there are six Poisson ratio values. However, only the three shown need to be entered, as the remainder are obtained from the expression:

Ej nij = Einji

Each individual Poisson ratio value must satisfy (nij < 0.5). For 2D anisotropic problems, you only need to enter the local x axis components. The local y is obtained by the right hand rule. Properties are only needed for directions 1 and 2, and plane 1-2 for axisymmetric and plane stress problems.

Non-Linear Isotropic Materials

If you choose the Non-Linear Isotropic Material type, the panel changes and prompts you to add two additional properties:

Initial Yield Stress (N/m2)

Hardening Parameters N/m2)

277


Material Property Inputs for Non-Linear Isotropic Materials

Stress Module Implementation-Model Setup and Solution-Volume Conditions-Piezoelectric Properties

To activate Piezoelectric Properties:

1. Ensure that the Stress box is activated in the PT tab.

2. Click the VC tab and from the VC Setting Mode menu, select Properties.

3. In the Model Explorer, select a volume with a Solid VC Type. An additional portion of the VC panel appears.

4. From the VC Setting Mode Properties menu, select Solid.

5. Click the Phys tab and enter Density - kg/m3

6. Click the Struct tab and from the Materials Type menu, select Linear.

7. Enter the following data:

Young's Modulus (modulus of elasticity) -N/m2

Poisson's Ratio - dimensionless

Expansion Coefficient (coefficient of thermal expansion) - m/m-K.

8. Click the Piezo tab and enter the following:

Dielectric Matrix - enter diagonal elements of 2x2 for 2D problems and enter 3x3 for 3D problems

Piezoelectric Coupling Matrix - enter 4x2 matrix (stress components and 2 geometry/material axes) for 2D or 2D axisymmetric problems and enter 6x3 matrix for 3D problems


Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Introduction

Click the Boundary Conditions [BC] tab to see the Boundary Conditions Panel. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or the BC Explorer.

The Stress Module is not supported by the thin wall or arbitrary interface boundary conditions. This does not mean that these types of boundary conditions cannot be used. It just means that they cannot be used within any solid volume condition regions where stress solution has been activated.

Stress Module boundary conditions only need to be given for boundary conditions that surround any solid region where stress calculation has been activated. This means that the only boundary condition types that need to have stress conditions applied are walls, rotating walls, and interfaces. Once one of these types of boundary conditions has been selected, all of the settings required for the Stress Module are located under the Stress tab and can be reached when the boundary condition setting mode is set to General.

For any wall, rotating wall, or interface there are four types of stress conditions, known as subtypes, that may be applied: Free, Prescribed Displacement, Loads, Symmetry, and Cyclic.

278

Stress Module

In addition to these, if contact analysis was specified in the Model Options section, a Contact Analysis checkbox will appear. If the chosen boundary is part of the contact model, either as a contact or a target face, this button should be activated and the type of face specified.

The Stress Boundary Conditions section includes:

Boundary Conditions-Free

Boundary Conditions-Prescribed Displacement

Boundary Conditions-Load


Boundary Conditions-Contact and Target Surfaces


Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Free

The Free boundary condition (the default) specifies a free surface with no constraints and no applied forces.

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Prescribed Displacement

The Prescribed Displacement boundary condition fixes the displacement of the specified boundary to the values or expressions that you enter into the fields for Delta X, Delta Y, Delta Z. There are two evaluation methods for the Prescribed Displacement subtype: Constant and Expression.

For the Constant evaluation method you must enter a real number. Then, the displacement value for the chosen boundary will be fixed to this constant value for the duration of the simulation. Setting the fixed displacement values to zero ensures that the selected boundary does not move.

If you choose the Expression evaluation method, you may enter a mathematical expression, specifying the displacement value as a function of the spatial variables (x, y, z) and/or time (t). The basic math functions: (sin, cos, tan, exp, +, -, *, /, **) are supported. In addition, the constant

PI is recognized as . All of the expression inputs are case insensitive.

For example, a displacement that varies with x-location may be input by specifying the subtype to be Prescribed Displacement, choosing the Expression evaluation method, and entering, for example:

0.0001*(X-1.0)

Boundary Conditions-Load

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Load-Introduction

The Load subtype allows forces to be applied to the selected boundaries. You can apply load options in any of the following combinations:

Applied Pressure

Implicit Pressure

Implicit Shear Stress

Spring Force

279


Point Load

Boundary Condition Load Subtype Options

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Load-Applied Pressure

The Applied Pressure subtype applies a fixed pressure to the specified boundary.

Load Subtype - Applied Pressure Option

There are two evaluation methods for applied pressure: Constant and Expression.

To specify applied pressure as a constant:

1. Under Applied Pressure, select the Constant option.

2. Enter a real number in the P field. The applied pressure for the boundary will be fixed to this constant value for the duration of the simulation.

To specify applied pressure as an expression:

1. Under Applied Pressure, select the Expression option.

2. Enter a mathematical expression as a function of the spatial variables (x,y,z) and /or time (t). The basic math function (sin,cos, tan, exp, +, -, *, /, **) is supported. The constant Pi

is recognized as . The expression inputs are NOT case sensitive. For example, a sinusoidal pressure force may be entered by specifying the subtype as Applied Pressure,

choosing the Expression evaluation method, and entering: 1000.0*sin(2.0* *T)

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Load-Implicit Pressure

280

Stress Module

The Implicit Pressure subtype specifies that the chosen boundary is coupled to another module and is to get its applied load from that module. This subtype is visible only when the Flow Module or the Electric Module is selected. Contributions from all modules will be used. For example, if the Flow and Electric modules are chosen in addition to the Stress Module, any boundaries where Implicit Pressure is specified will use the calculated fluid pressure and an effective electrostatic pressure as the applied pressure.

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Load-Implicit Shear Stress

The Implicit Shear Stress subtype includes forces due to fluid shear stresses at a fluid-solid interface.

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Load-Spring Force

The Spring Force subtype models a boundary that is assumed to be connected to a rigid plane by a spring. If you choose this subtype, an additional panel appears requiring:

Constant - enables you to specify the Spring Constant (k).

Boundary Condition - Load - Spring Force Option

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Load-Point Load

The Point Load subtype specifies the location and force components of one or more point loads. These loads are applied to the closest node on the selected surface.

281


Boundary Condition - Load - Point Load Option

To specify constant point load values:

1. Select the Point Load checkbox to activate it.

2. In the Total Point Loads field enter the number of points that you want to specify and click OK. The Current Point Load field is set to 1 indicating that parameters for the first point can now be set.

3. Select Constant or Parametric from the menu. To specify constant points, enter the X-Coordinate, Y-Coordinate, Fx, and Fy. Use the arrow located to the right of the Current Point Load field to scroll through each next point that you entered in the Total Point Loads field. If Parametric was chose, you must define the parametric input for Fx and Fy .

To define parametric input:

1. Select the Define button to the right of the Fx parameter

The Parametric Input window appears: See Tools Menu-Parametric Input for details

282

Stress Module

2. From the Parametric Input dialog click Add to create a variable. The variable appears in

the list window and two type in fields appear. The field on the left enables you enter a new name for the variable and the field on the right enables you to enter the value for the variable.

283


3. To rename the variable, enter a new name in the type in field located at the bottom left of the variable list.

4. To specify the value for the variable, use the functions provided at the bottom of the window and/or type the desired formula directly in the value field

5. To add more variables, click the Add button. If you make a mistake, use the Delete button to delete a specific variable or use the Reset button to clear the Parametric Input dialog.

6. Click Apply to accept changes and OK to save.

7. In a similar manner, define the Fy parametric input.

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Symmetry

284

Stress Module

If a Symmetry type boundary condition borders a solid region where stress has been activated (instead of a wall or interface type), that information is passed to the Stress Module that then treats it as a symmetry. No further input is required.

However, there are times when a structural symmetry condition is desired on a boundary which is not modeled as symmetric for the other modules (for example you may want to fix a temperature at this boundary). You can specify a structural symmetry condition on a wall or interface boundary by choosing the Symmetry subtype.

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Contact and Target Surfaces

If you select Contact Analysis in the Model Options section, a Contact checkbox appears under the Stress boundary condition panel. Selecting this box will cause a surface type box to appear under the SubType box. You can select the surface to be either a contact or a target surface.

Contact and target surfaces are defined in pairs and are matched using a Pair Name. All contact surfaces with a certain pair name can potentially come into contact with all target surfaces of the same name. A contact surface group may be a member of only one pair, but a target surface group may be a member of up to five different pairs.

For contact surfaces, you are prompted to enter the pair name for that surface. For target surfaces, you are prompted for pair names and contact gap.

Contact Target Surface Parameters

As mentioned, the target surface may be a member of several pair sets.

To define target surface sets:

1. Click the Specify Contact Target button. The Contact Target Pair window appears.

285


2. In the Total Pairs field, enter the total number of pairs for the target surface then click on the OK button.

3. In the Pair Name box, enter a name for each pair. Each target pair name should have a corresponding pair name for a contact surface.

4. Click the Close button to close the Contact Target Pair dialog.

A Contact Gap is specified for each target surface. This is the contact/target gap size at which contact is assumed to occur. This is needed when there is a fluid region between the contact and target surfaces. If you specify a non-zero contact gap, the fluid cells will be squeezed into that distance between the two surfaces, rather than going to zero and producing zero or negative volumes.

Stress Module Implementation-Model Setup and Solution-Boundary Conditions-Cyclic

Cyclic or periodic boundary conditions for Stress are implemented using the following constraint equations:

Master (m) Slave (s)

Displacement

Displacement

Normal

Normal

Tangential Vectors ,

Tangential Vectors ,

286

Stress Module

-ve sign, since normals are opposite to each other for periodic faces (9-23)

(9-24)

(only 3D) (9-25)

For setup of cyclic/periodic BC, see Cyclic Boundary Conditions.

Stress Module Implementation-Model Setup and Solution-Initial Conditions

Click the Initial Conditions [IC] tab to see the Initial Conditions Panel. The Initial Condition panel appears:

Initial Conditions Panel

287


You can specify Initial Conditions as constant values or have them read from a previously obtained solution file. If you specify constant values, the Stress Module assumes zero initial values for displacement, velocity, and acceleration for a non-restart run. You can also define initial temperature and residual stress.

The Initial Temperature is the temperature value for thermoelastic analyses in which the heat module is not activated (i.e. the temperatures are constant rather than calculated from the energy equation). For each volume in the structural model, the temperature difference used to calculate thermoelastic strain is the difference between this initial temperature and the reference temperature of the volume (specified in the VC panel).

Initial (or residual) stresses are also specified in this section. Axial stresses in the global x,y, and z directions are input in the respective boxes. These stresses are defined as positive in tension, negative in compression.


Stress Module Implementation-Model Setup and Solution-Solver Control-Introduction

Click the Solver Control [SC] tab to see the Solver Control Panel. The Solver Control panel accesses settings that control the numerical aspects of the CFD-ACE-Solver. The Stress Solver Control section includes:

Solver Control-Iterations



Stress Module Implementation-Model Setup and Solution-Solver Control-Iterations

The Solver Control Iterations tab enables you to set the maximum number of iterations and convergence criteria. If you have chosen a nonlinear structural option (i.e. large deformation, contact analysis) in the Model Options panel, you can enter the number of iterations of the Newton-Raphson algorithm in the Iteration field located under the Stress section of the Iter tab of the SC panel. This iteration value is separate from the number of iterations of the main solver (specified in the Max. Iterations field in the Shared section), since the Stress Module converges to a true structural solution each time the solver is called (otherwise, a totally unrealistic geometry could be sent back to the other modules).

288

Stress Module

Solver Control - Iteration Tab - Stress Parameters

Coupling Frequency Iterations

If you also activated the Grid Deformation Module, the Coupling Frequency field appears in the Stress section of the Iter tab. This variable specifies the frequency, in iterations, that Femstress is to be called to update the grid. Femstress will always be called at the end of each time step (for transient simulations), commonly known as one-way coupling, or at the end of the solution (for a steady state simulation). Thus, if a steady state case converged in 22 iterations and the Coupling Frequency was specified as 5, Femstress will be called five times (that is, at iterations 5, 10, 15, 20, and 22).

The default value is set to a very large number (100000) to treat the problem as one-way coupling. For most FSI problems, the coupling frequency needs to be smaller than the maximum iterations.

Stress Module Implementation-Model Setup and Solution-Solver Control-Solver Selection

The Solver Control Solvers tab enables you to select the linear equation solver to be used for the solution of the stress equations from the Stress section. The options are:

Automatic - This option automatically chooses the solver to use for the simulation based on the problem size. If CFD-ACE+ chooses the CGS solver, you must specify the number of sweeps and convergence criterion to stop the iterative process

Direct Solver - This is a linear solver that is recommended for any problems involving shell elements, since shell elements result in very ill-conditioned systems (Benzi, 1997).

CGS Solver - The Conjugate Gradient Solver is an iterative linear solver. For this solver, you must specify the number of sweeps and convergence criterion to stop the iterative process.

289


SC Panel - Solvers Tab - Stress Equation Options

Stress Module Implementation-Model Setup and Solution-Solver Control-Under Relaxation Parameters

The Solver Control Relax tab enables you to select the amount of under-relaxation to be applied for each of the dependent (solved) and auxiliary variables used for the stress equations.

The default values for all of the under relaxation settings will often be sufficient. In some cases, these settings will have to be changed, usually by increasing the amount of under-relaxation that is applied. There are no general rules for these settings and only experience can be a guide. There are two possible Stress Module related inputs:

FEM Contact - The contact model used in the Stress Module requires some linear relaxation for optimal convergence. The default value of 0.5 will usually suffice, although values as low as 0.1 may be needed in some cases.

Grid Motion - If you have activated the Grid Deformation Module, then the calculated grid deformation is sent back to the Flow or other modules. Relaxing this motion by checking the Grid Motion box will give the flow field time to adjust. Typical values will range from 0.05 to 1.0, depending on the severity of the deformation and the coupling between the flow and stress.

290

Stress Module

Solver Controls - Relax Tab with Grid Motion Option Available

Stress Module Implementation-Model Setup and Solution-Solver Control-Adv

The Solver Control Adv tab contains two sections:

Shared - When checked, adds buffered output (what is this and what does it share with?)

Stress - Adds Gaussian Integration Points.

o First Order

o Second Order

291


Solver Control - Advanced Tab


Stress Module Implementation-Model Setup and Solution-Output-Introduction

The Output (Out) Panel enables you to modify output location, set output frequency and type, and select user subroutines. The Output section includes:

Output-Output

Output-Print

Output-Graph

Output-Monitor

292

Stress Module

Output Panel

Stress Module Implementation-Model Setup and Solution-Output-Output

The Output (Out) Panel's Output tab enables you to modify output location, set output time step and frequency, and select user subroutines.

293


Output Panel

Transient Results

The Transient Results section contains an Output Location menu with three options:

Unique Filename - sends the results to a unique filename

Unique Simulation - sends the results to a unique simulation

Replace Simulation - sends the results to replace and existing simulation

Output Frequency

The Output Frequency section contains:

Constant Time Step -

Starting Time Step - enables you to enter a starting time step for the output

Ending Time Step - enables you to enter a ending time step for the output

Time Step Frequency - enables you to generate one or more time step frequencies

User Defined Output

The User Defined Output section contains three user subroutines:

User Sub(uout) - When activated, this subroutine will be called from various points in the solution cycle.

User Sub(uread_dtf) - When activated, this subroutine will be called from solver at the beginning of simulation, when solver is reading the DTF file.

294

Stress Module

User Sub(uwrite_dtf) - When activated, this subroutine will be called from solver at the end of simulation, when solver is writing the DTF file. At this point, you should not do anything except write to the DTF file.

Mode Shape Output

If the Modal Analysis Model has been activated (see Model Options-Stress-Modal Analysis for details) the Mode Shape Output section appears on the Output tab. When you select this output option, the first N modes, where N is the number of requested modes in the Model Output section, are written to separate DTF files. These files follow the same numbering format as transient data files (that is, the first mode shape will be written to Model.0001.DTF, the second mode shape to Model.0002.DTF, etc).

The Mode Shape Output option opens the Maximum Plot Displacement section of the Output panel. It has two Evaluation Method options:

User Specified -Use this option to enter a value in the Absolute Disp. (displacement) field. This value represents the magnitude of the maximum deformation difference (that is, maximum displacement minus minimum displacement). For example, if you specify a value of 1.0, a mode shape with equal positive and negative displacements will range in value from -0.5 to 0.5.

Code Calculated - Use this option if you do not know a valid value for the Mode Shape Output parameter. CFD-ACE+ chooses the displacement based on the size of the structure. If these displacements are too large or too small, you can use the Scale Factor field to adjust the value.

Stress Module Implementation-Model Setup and Solution-Output-Print

The Output (Out) Panel's Output tab enables you to print specific portions of your solution. This panel changes to include additional print options if other modules have been activated. For the Stress module only, it contains two sections:

Shared - The Shared section contains the following options:

o BC Integral Output

o Diagnostics

o Set Residual Frequency

Stress - When you activate the Stress Module, it outputs a reaction force summary in addition to the general printed output options. The Reaction Forces option prints the reaction forces at each fixed node of the structural model.

295


Output Panel - Print Tab

Stress Module Implementation-Model Setup and Solution-Output-Graphic

The Output (Out) Panel's Graphic tab enables you to select the variables to output to the graphics file (modelname.DTF). These variables will then be available in CFD-VIEW.

Output Panel - Graphic Tab

Shared

The Density option [need description]

Stress

296

Stress Module

The Stress graphical output variables are:

Displacement - [need description] (Units - m)

Cartesian Stress Tensor - [need description] (Units - N/m2)

Cartesian Strain Tensor - [need description] (Units - none)

Principal Stress - Provides the maximum and minimum normal stresses, the maximum shear stress, and the VonMises stress. (Units - N/m2)

Principal Stress Direction - Provides a vector that shows the direction of the maximum and minimum principal stresses. This may be viewed as a vector in CFD-VIEW. The sign of the vector is undefined (for example, a point in tension is pulled in each direction). An arbitrary condition is used to give the vectors a sign. (Units - none)

Principal Strain - [need description] (Units - none)

Reaction Forces - Provides the resulting x,y,z direction forces at all fixed boundaries. (Units - N)

Stress Module Implementation-Model Setup and Solution-Output-Monitor

The Monitor tab enables you to specify one or more locations where solution variables will be printed. For more information, see Overview-Monitor Points. All Graphical variables are available under monitor point output.

Output Panel - Monitor Tab - Monitor Point Definition and Shared Sections

297


Output Panel - Monitor Tab - Stress Section

Not shown are X-Reaction Force and Y-Reaction Force

Stress Module Implementation-Post-Processing

Use CFD-VIEW to post-process the solutions. When you activate the Stress Module, you will want to view structural displacements (Cartesian values) and principal stresses. By definition, the Normal Stress values are defined as positive in tension and negative in compression. The following table shows a complete list of post processing variables available (depends on the Graphical Output selections you have selected) as a result of using the Stress Module:



TauMax Maximum shear stress N/m2

298

Stress Module

VonMises VonMises Stress N/m2

Stressxx Cartesian stress component xx N/m2

Stressyy Cartesian stress component yy N/m2

Stresszz Cartesian stress component zz N/m2

Stressxy Cartesian stress component xy N/m2

Stressxz Cartesian stress component xz N/m2

Stressyz Cartesian stress component yz N/m2

Strainxx Cartesian strain component xx -

Strainyy Cartesian strain component yy -

Strainzz Cartesian strain component zz -

Strainxy Cartesian strain component xy -

Strainxz Cartesian strain component xz -

Strainyz Cartesian strain component yz -

StrainMax Maximum principal strain -

StrainMin Minimum principal strain -

Shearmax Maximum shear strain -

ShearMin Minimum shear strain -

Fx x-direction reaction force N

Fy y-direction reaction force N

Fz z-direction reaction force N

Displacements are solved at the nodes of the element, whereas the stresses are post-processed from the displacement field and are calculated at the Gaussian integration points internal to the elements. The stresses are calculated from displacement derivatives, and thus are one order less accurate than the displacements. Also, since the stresses are calculated at internal points and interpolated to the nodes, the stress value at exterior surfaces actually represents the stress at the integration points inside that surface element.

Stress Module Frequently Asked Questions

What is the Cartesian stress tensor?

At each point, we have the stress tensor ij, where i refers to the face and j refers to the direction:

299


The nine components ij make up the Cartesian stress tensor. The stress tensor is symmetric, i.e. ij=ji, so there are six independent components of the Cartesian Stress Tensor. At each face, the three components of stress on that face sum vectorally to a force (per unit area) on that face,

(9-26)

The full stress tensor can combine to create a stress in a general direction:

A simple force balance will give:

(9-27)

where is simply the component representation of .

From equation 9-26 and equation 9-27:

300

Stress Module

(9-28)

or

(9-29)

The force (stress) vector is not necessarily normal to face.

What are principal stresses?

Consider a plane where the normal component of is an extremum. Some math results in:

1. This is an Eigenvalue problem with three solutions.

2. These planes are planes where is in face normal to the face.

3. Therefore, the shear stress on these planes is zero.

Planes where is maximum give ax

Planes where is minimum give min

The maximum shear stress is given by:

(9-30)

max acts on planes bisecting the planes of max and min.

The VonMises stress is related to the distortional energy of the body (as opposed to the hydrostatic) and is given by

(9-31)

where 1, 2, 3 are the three principal stresses.

I notice that sometimes the displacement values I see in the contour plots do not match the actual displacement of the grid. Why is that?

This may happen for coupled fluid/solid problems with a grid relaxation parameter less than 1.0, which are not fully converged. The grid relaxation is used for problems with very large deformations or problems with initially very large pressures, where we do not want to send the full grid change back to the fluid solution.

Wouldn’t that mean that the two solvers are almost solving two "independent" problems anytime there is a case where the grid motion relaxation is less than 1.0?

301


The problems are not totally independent, since the grids are related. This semi-independence fits in with the overall sequential and explicit coupling of the stress and flow solutions, in which each solution is obtained with information passed from the other solution until convergence is obtained. The merging of the two grids is part of the convergence process.

For problems that are very difficult to converge, grid relaxation values much less than 1.0 may be needed. For these problems, are any modifications made to account for that fact that the stress and flow grids may differ?

No type of modification is done to account for the different grids. The thought is that they should only differ substantially in the early stages when the fluid forces also tend to differ substantially from the final solution. If these fluid forces are way off, we typically do not want to use the full displacement that these incorrect forces would cause. As the solution converges the differences in the grids goes away.

What other methods of obtaining convergence for FSI problems can be used to avoid using extremely small values of grid relaxation?

Convergence is often problem-specific, but here are some suggestions that should help in most problems:

If the problem is transient, using a smaller time step (if feasible) will help. If the problem is steady, sometimes running it as a transient (to reach steady-state) will help convergence.

Linear relaxation on pressure helps to moderate the pressure fluctuations seen by the stress solver, reducing the displacement fluctuations and aiding convergence.

Limiting the pressure values is often useful. Sometimes the pressure field will see unrealistic values during the first few iterations, which will produce unrealistic deformations (which then feed back into the flow solver). Often the user will know the approximate maximum and minimum pressure values of the final solution, which can be used to set pressure limits. With these limits, the non-physical pressure values will not be sent to the stress solver.

The coupling frequency option can be used to allow the flow field to develop before the stress solver is called for the first time. For example, with a coupling frequency of 5, there will be 5 flow iterations before the first call to the stress solver, which would give time for the pressure and viscous forces in the flow field to reach more realistic values.

What is Femstress?

FEMSTRESS is the old name for the Stress Module. The Stress Module is a Finite Element based Structural Analysis Module of CFD-ACE+. It can be used in stand-alone mode or coupled with the following types of problems:

Flow (Coupled Flow and Stress/Strain)

Heat Transfer (Stress/Strain due to Thermal + Fluid Effects)

Electrostatics (Due to Electrostatic Forces)

Piezoelectric (Due to Electrostatic Forces in piezoelectric Materials)

What element types are supported?

2D Elements

Shapes: Triangles, Quadrilaterals

Formulation: Plane Stress, Plane Strain, Axisymmetric

3D Elements

Shapes

o Tetrahedral, Prismatic, Pyramidal

302

Stress Module

Standard Brick Elements (First and Second Order)

o Hexahedral

Standard Brick Elements (First and Second Order)

Enhanced Brick Elements (For bending dominated and incompressible problems) (First Order)

Solid Shells (First Order)

Which element type should I use?

First order elements are the most robust and efficient and can be used for many simulations.

If you have a bending dominated problem (like the bending of a plate or beam) then second order elements (or enhanced first order bricks) should be used.

If (nearly) incompressible behavior is present (e.g., in linear elastic materials with Poisson's ratio greater than 0.49 or nonlinear elastoplastic materials) then enhanced first order elements or second order elements will perform better than standard first order elements.

For (thin) shell structure analysis, solid-shell element is more accurate than standard first or second order elements.

How do I use shell elements?

Problem: User cannot figure out how to activate Shell Surface element type.

Solution: Specify one side of thickness direction as Shell Surface, under BC type.

Another Common Problem:

303


You created a grid with multiple cells in thickness direction

You must have only 1 cell in the thickness direction to use Solid Shell elements

What value do I set for the contact gap?

Set a value of about 1 or 2 orders smaller than the unsqueezed grid resolution. If you use 3 or 4 orders smaller, the grid-cell aspect ratio will be so high that convergence problems may result.

Stress is on but no stress is predicted.

You may have forgotten to click Activate Stress. Once you have activated the Stress Module, you must also change the VC Setting Mode to Stress and click Activate Stress in each individual VC where Stress is to be calculated.

Help! I Have Negative Volumes! (FSI-related problems)

Output File Snippet:

--- Negative Volumes Encountered

Cell No. = 68

X Y = 1.13589721737274 0.485607586491270

Volume = -8.681406085241693E-004

********************************************************************

Error: The new grid has negative volumes, which will cause failure of the solver. The negative volume may be identified in CFD-VIEW by negative values of the scalar variable "ng_Vol" in DTF file name =Shear_FSI1_negative.00001.DTF in sim = 1

********************************************************************

CPU Time at the end of = 1 time steps.

End of Output Elapsed Time= 1.894724E+01 Delta-time= 8.392066E+00

--- Negative Volumes Encountered

304

Stress Module

Negative volumes are generated when the structure deforms so much that the fluid-grid cells get highly distorted. Probable causes are large implicit pressures or divergence/non-convergence of nonlinear FEM iterations.

You may tend to miss the FEM.RSL file below file because it is not accessible through the GUI, so you will not usually catch this problem yourself.

FEM.RSL File Snippet

# ACE_ITER FEM_ITER Node DOF x y z Max dphi Energy Norm

#FEMSTRESS Convergence at outer iteration Number 1 of time step 1

1 1 323 1 -1.3000E-04 3.4616E-03 0.0000E+00 -2.0731E-08 5.0813E-08

1 2 240 2 5.0000E-05 7.3500E-03 0.0000E+00 2.8477E-14 1.3353E-17

#FEMSTRESS nonlinear algorithm converged at iteration Number = 2 with DPMAX = 2.8477E-14

To repair negative volumes:

Change the inlet boundary condition for velocity or pressure so that implicit force on the solid is not very high.

Change the pressure and grid relaxation parameters.

If transient, reduce the time step size by one or two orders. Restart from the last saved time step DTF file before negative volumes, with a much smaller time step size.

If contact, refine the grid to make the contact surfaces more smooth. Problem is cross-over due to very different resolution (5:1 or more).

SPC file

Stress-Examples

Stress Module Examples

The Oil Flow through a Compliant Orifice Tutorial uses the Stress Module with one or more other Modules.

Stress-Examples-Demo Problems

Stress Module Examples-Stress Concentration

Create two circles centered at the origin, of radii 0.2 and 0.4 m, and split each point at a parametric value of 0.25 (i.e. at x=0.0). Put in points C (1,0,0), F (0,1,0) and G (1,1,0). Delete all but the first quadrant of the circles, and connect the points with lines so that your geometry looks as shown below.

305


The inner region (ABED) will be meshed with a structured grid, and the outer region (BCGFE) with an unstructured grid. Create edges on the four curves of the inner domain (AB, BE, ED, DA), with 11 points along the straight lines and 15 points along the curved lines. Then, create a face on those edges.

On the outer region, create an edges along the x and y axes (BC and EF, respectively) with 7 points, using the power law distribution with a factor of 1.7. On the other two lines (CG and FG), create edges with 6 points, equally spaced.

On the inner region, create a face from the four edges, and a 2D block from the face. Create a loop on the edges of the outer domain, use the Tools->Utilities option to create a 2D domain from the loop, and then create a triangular grid on that 2D domain. Your grid should now look like that shown below.

Stress Concentration Geometry

306

Stress Module

Stress Concentration Grid

In the BC/VC editor, set both the face and the loop to solid domains. Specify the edges along the x and y axes as symmetry faces, and set the other boundaries to walls. Then save this as a DTF file.

Read this DTF file into CFD-ACE+. When it is read in the unstructured grid and the structured grid should be represented as a face (i.e. without the grid shown).

In the Problem Type/Modules section, activate the Stress module. The parameters under the Global section stay at their default values. Under the Model Definition section, choose plane stress as the Geometry type, and keep all other parameters at their default values.

In the Boundary Condition panel, apply a fixed pressure of 10,000 Pa to the outer wall surfaces, and leave the inner surface to the default of Free. The constraints will come from the symmetry surfaces set in CFD-GEOM.

In the Volume Condition section, set the properties of both domains to that for steel (E=2.e+11 Pa, =0.3). The density and coefficient of thermal expansion are not relevant for this analysis. In the General section of Volume Conditions, set Equations and Stress Calculation for each volume.

In the Solution Control section, click on Cartesian stress tensor, to get these values written to the DTF file for post processing. We are now ready to run the problem.

This problem should run quickly. After it runs, read the DTF file into CFD-VIEW, and look at contours of Sigmaxx and Sigmayy. The stress values are positive in tension and negative in compression. The applied pressure on the outer surfaces will cause a compressive stress. With a stress intensity factor of 2 for this case, the minimum value of these stresses should be near 20,000.

Stress Module Examples-Hoop Stress

The figure below shows the grid used for the Hoop Stress study. The geometry consists of an infinitely long thick-walled cylinder of inner radius 1 m and outer radius 1.5 m. A constant pressure of 10,000 Pa is applied on the inner surface. The geometry is modeled with 8-noded (first order) brick elements, clustered near the inner surface, as shown. A 1/4 sector of the cylinder was modeled, with symmetry boundary conditions on the circumferential and axial faces. The relevant properties are:

E = 200 Gpa

307


= 0.30

Geometry for the Stress Analysis

The following figure shows the radial displacement contours (obtained using the Calculator Tool in CFD-VIEW). Analytical values for the inner and outer displacement are 1.378e-8m and 1.092e-8 m, respectively.

Radial Displacement Contours

Stress Module Examples-Large Deflection

The figure below shows the initial and final configurations for the large displacement solution of a cantilever beam deforming under its own weight. The beam has a length of 3 m and a length to thickness ratio of 30. The beam properties are:

E = 23400 Pa

= 0.0,

= 1.0 Kg/m3

Gravitational loading was taken such that the non-dimensional parameter K= WL3/EI was equal to 20, where W is the load per unit length. The table summarizes the analytical and numerical values for normalized tip deflection (horizontal and vertical).

308

Stress Module

Large Deflection of a Cantilevered Beam

Normalized Tip Deflections

h/L V/L

Analytical 0.445 0.830

CFD-ACE-GUI 0.443 0.832

Stress-Examples-Validation Cases

Stress Module Examples-Stress Concentration in a Circular Cylinder

Reference

Roark and Young, Formulas for Stress and Strain, Page 600.

Elements

4 noded tetrahedrals

Properties

E=200 Gpa, =0.30

Details

The geometry consists of a circular beam with a diameter change. The large radius is 2.0 m, the small radius is 1.0 m, and the fillet radius is 1.0 m. The large diameter end is fixed and a tensile force is applied to the other end. A 1/8 sector modeled with symmetry BC’s. The grid and stress contours are shown below.

309


Stress Concentration in a Circular Cylinder

Results

Maximum stress concentration:

Roark & Young: 1.33

Stress Module: 1.33

Stress Module Examples-Thermoelastic Deformation of a Cylinder

Reference

Boley and Weiner, Theory of Thermal Stresses, pg 290.

Elements

4-noded quadrilaterals in axisymmetry.

Properties

E=20 Gpa, = 0.0, = 5.0e-5 K-1

Details

310

Stress Module

A cylinder of inner radius ri = 0.5m and outer radius ro = 2.5 m was modeled as a 2D axisymmetrical geometry. The inner surface was held at a temperature of 200K and the outer surface at 100K, relative to the unstressed temperature value. A value of =0 was used because the sides of the cylinder were modeled in the Stress Module as symmetric walls (so the problem would not be unconstrained), and thus no axial displacement was allowed. Using =0 uncouples the radial and axial displacements to allow comparison with the analytical values. The grid and results are shown below. The radial displacement and circumferential stress are given by:

(9-32)

(9-33)

Results Table

uinner(m) uouter(m) sinner(Mpa) outer(Mpa)

Analytical 0.00317 0.0159 -73.10 26.90

CFD-ACE-GUI 0.00318 0.0159 -68.78 26.62

311


Thermoelastic Analysis in a Cylinder

Stress Module References

Bathe, K., and Wilson, E.L., Numerical Methods in Finite Element Analysis. Prentice-Hall, 1976.

Belytschko, T., and Mindle, Walu, "The Treatment of Damping" Transient Computation in Damping Applications for Vibration Control. P.J. Torvik, ed., ASME AMD, Vol. 38, 1980, pp. 123-132.

Benzi, M., Kouhia, R., and Tuma, M., An Assessment of Some Preconditioning Techniques in Shell Problems. Los Alamos National Laboratory Technical Report LA-UR-97-03892, 1997.

Cook, R.D., Malkus, D.S., Plesha, M.E., Concepts and Applications of Finite Element Analysis. 3rd Edition, John Wiley and Sons, 1989.

Newmark, N.M., “A Method of Computation for Structural Dynamics.” JEMDiv, 85.EM3 (1959), 67-94.

Roark, R.J., and Young, W.C., Formulas for Stress and Strain. McGraw-Hill, 1975.

Saad, Y., Iterative Methods for Sparse Linear Systems. Boston: PWS, 1996.

Simo, J.C., Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory, Computer Methods in Applied Mechanics and Engineering, 99, pp.61-112, 1992.

Zienkiewicz, O. C., The Finite Element Method in Engineering Science. McGraw-Hill, 1971.

312

Stress Module

313

Optimal solid shells for nonlinear analyses of multilayer composites. Part I: Statics '', L. Vu-Quoc and X.G. Tan, Computer Methods in Applied Mechanics and Engineering, Vol.192, pp. 975-1016, 2003.

Optimal solid shells for nonlinear analyses of multilayer composites. Part II: Dynamics '', L. Vu-Quoc and X.G. Tan, Computer Methods in Applied Mechanics and Engineering, Vol.192, pp. 1017-1059, 2003.

Appendix A - Post Processing Variables

Appendix A Post-Processing Variables

Variable Description Units Module

D_AnalyteName Diffusivity m2/s BioChem

Surfcon_AnalyteName Surface Concentration moles/m2 BioChem

AnalyteName Analyte Concentration M BioChem

Irrcon_AnalyteName Irreversible Analyte Concentration

M BioChem

Reaction_Rate Generation Rate of Product/consumption Rate of Substrate

moles/m2/sec BioChem

Wall_conc_AnalyteName Wall concentration M BioChem

MassFr Mass fraction - Cavitation

Total_Volume_Fraction Total volume fraction - Cavitation

Vapor_Volume_Fraction Vapor volume fraction - Cavitation

Nox_Rate Nox production rate - Chemistry

Progress Progress Variable - Chemistry

React_Rate Reaction Rate - Chemistry

Species name Species Mass fraction - Chemistry

efieldx Electric Field, x-component N/C Electric

efieldxi Electric Field, imaginary x-component

N/C Electric

efieldy Electric Field, y-component N/C Electric

efieldyi Electric Field, imaginary y-component

N/C Electric

efieldz Electric Field, z-component N/C Electric

efieldzi Electric Field, imaginary z-component

N/C Electric

el_epsr Dielectric constant or relative permittivity

- Electric

el_pot Electric Potential Volt Electric

epsr_x_En Weighted electric field normal N/C Electric

314


Jcx_i Conduction current density, imaginary x-component

A/m2 Electric

Jcx_r Conduction current density, real x-component

A/m2 Electric

Jcy_i Conduction current density, imaginary y-component

A/m2 Electric

Jcy_r Conduction current density, real y-component

A/m2 Electric

Jcz_i Conduction current density, imaginary z-component

A/m2 Electric

Jcz_r Conduction current density, real z-component

A/m2 Electric

mu_r Relative Permeability - Electric

pestat Electrostatic pressure N/m2 Electric

Qsurf Surface charge density C/m2 Electric

Qvol Volume Charge density C/m3 Electric

sig_i Imaginary part of conductivity -1m-1 Electric

sig_r Real part of conductivity -1m-1 Electric

vfx_estat Virtual electric force, x-component

N/m3 Electric

vfy_estat Virtual electric force, y-component

N/m3 Electric

vfz_estat Virtual electric force, z-component

N/m3 Electric

Mach Mach Number - Flow

P Static Pressure N/m2 Flow

P_tot Total Pressure N/m2 Flow

RHO Density kg/m3 Flow

STRAIN Strain - Flow

Stream_Function Stream Function m2/s Flow

U X-direction Velocity m/s Flow

V Y-direction velocity m/s Flow

Vorticity_Crit Vorticity Criteria - Flow

Visc Effective Viscosity kg/m/s Flow

Visclam Laminar Viscosity kg/m/s Flow

W Z-direction Velocity m/s Flow

315


COND Conductivity W/m-K Heat

CONDY Y-direction Conductivity W/m-K Heat

CONDZ Z-direction Conductivity W/m-K Heat

CP Specific Heat J/kg-K Heat

H0 Total Enthalpy m2/s2 Heat

T Temperature K Heat

T_TOT Total Temperature K Heat

Wall_Heat_Flux Wall heat Flux W/m2 Heat

Wall_Rad_Flux Wall radiative flux W/m2 Heat

Ax_r Magnetic Vector Potential Real x Component

Wb/m Magnetic

Ax_i Magnetic Vector Potential Imaginary x Component

Wb/m Magnetic

Ay_r Magnetic Vector Potential Real y Component

Wb/m Magnetic

Ay_i Magnetic Vector Potential Imaginary y Component

Wb/m Magnetic

Az_r Magnetic Vector Potential Real z Component

Wb/m Magnetic

Az_i Magnetic Vector Potential Imaginary z Component

Wb/m Magnetic

J_eddy Eddy current A/m2 Magnetic

Power_dissipation Power dissipation W/m3 Magnetic

Ex_i RF electric field, imaginary-x component

N/C Magnetic

Ey_i RF electric field, imaginary-y component

N/C Magnetic

Ez_i RF electric field, imaginary-z component

N/C Magnetic

|E|rf Magnitude of rf electric field N/C Magnetic

Bx_r Magnetic field real-x component N/A-M Magnetic

By_r Magnetic field real-y component N/A-M Magnetic

Bz_r Magnetic field real-z component N/A-M Magnetic

Bx_i Magnetic field imaginary-x component

N/A-M Magnetic

316


By_i Magnetic field imaginary-y component

N/A-M Magnetic

Bz_i Magnetic field imaginary-z component

N/A-M Magnetic

vfx_mag Virtual magnetic force, x-component

N Magnetic

vfy_mag Virtual magnetic force, y-component

N Magnetic

vfz_mag Virtual magnetic force, z-component

N Magnetic

fx_mag Magnetic force, x-component N Magnetic

fy_mag Magnetic force, y-component N Magnetic

fz_mag Magnetic force, z-component N Magnetic

Te Electron temperature eV Plasma

Ne Electron number density 1/m3 Plasma

nu Collision frequency s-1 Plasma

Jx_Ne Current density, x-component A/m2 Plasma

Jy_Ne Current density, y-component A/m2 Plasma

Jz_Ne Current density, z-component A/m2 Plasma

Ne_avg Average Electron number density m-3 Plasma

Te_avg Average electron temperature eV Plasma

Pwr_avg Average power W/m3 Plasma

Mob_e Electron mobility m2/(V-s) Plasma

dep_avg Deposition rate kg/m2s Plasma

Power Power W/m3 Plasma

E_x x component of ambipolar field V/m Plasma

E_y y component of ambipolar field V/m Plasma

E_z z component of ambipolar field V/m Plasma

D_ScalarName Scalar Diffusion Coefficient kg/m-s Scalar

ScalarName Scalar Name - Scalar

spr_srcu u-momentum source term kg m/s2 Spray

spr_srcv v-momentum source term kg m/s2 Spray

spr_srcw w-momentum source term kg m/s2 Spray

spr_srch enthalpy source term J/s Spray

317


spr_srcm mass source term kg/s Spray

spr_volfrac spray volume fraction - Spray

spr_disp particle concentration particles/m3 Spray

TauMax Maximum shears stress N/m2 Stress

Sigmazz Cartesian stress component zz N/m2 Stress

VonMises VonMises Stress N/m2 Stress

Sigmaxx Cartesian stress component xx N/m2 Stress

Sigmaxy Cartesian stress component xy N/m2 Stress

Sigmaxz Cartesian stress component xz N/m2 Stress

Epsxx Cartesian strain component xx - Stress

Epsyy Cartesian strain component yy - Stress

Epszz Cartesian strain component zz - Stress

Epsxy Cartesian strain component xy - Stress

Epsxz Cartesian strain component xz - Stress

Epsyz Cartesian strain component yz - Stress

EpsMax Maximum principal strain - Stress

EpsMin Minimum principal strain - Stress

Shearmax Maximum shear strain - Stress

ShearMin Minimum shear strain - Stress

Fx x-direction reaction force N Stress

Fy y-direction reaction force N Stress

Fz z-direction reaction force N Stress

D Dissipation Rate( model) Specific rate of dissipation ( model)

m2/s3 s-1

Turbulence

K Kinetic energy m2/s2 Turbulence

VIS_T Turbulent Viscosity kg/m-s Turbulence

YPLUS Yplus values - Turbulence

U2 X-direction velocity of 2nd fluid m/s Two Fluid

V2 Y-direction velocity of 2nd fluid m/s Two Fluid

318


319

W2 Z-direction velocity of 2nd fluid m/s Two Fluid

RHO2 Density of 2nd fluid kg/m3 Two Fluid

H2 Enthalpy of 2nd fluid kg-m2/s2 Two Fluid

Visc2 Viscosity of 2nd fluid kg/m-s Two Fluid

T2 Temperature of 2nd fluid K Two Fluid

Alpha Volume fraction of 2nd fluid - Two Fluid

Stream2 Stream Function of 2nd fluid - Two Fluid

LiqVOF Volume fraction of 2nd fluid - VOF

VOFOld Volume fraction at previous time step

- VOF

Rho_1 Density of fluid 1 kg/m3 VOF

Rho_2 Density of 2nd fluid kg/m3 VOF

curvtur curvature m-1 VOF

Index

2 2D axisymmetric ...... 16, 105, 154, 176, 200,

233, 244, 257, 269, 282 2D axisymmetrical ...................................314 2D Elements ............................................303 2D Hydrofoil Simulation ...........................227 2D-axisymmetric ........................................16 2nd Order Wall Global ...............................24 3 3D Elements ............................................303 A Absorption Coefficient .....................202, 209 absorption/emission.................................194 Actuators....................................................54 Adds Gaussian Integration Points ...........295 Adiabatic ....................................................67 Adiabatic Option ........................................67 Advanced Tab..........................................295 Air Injection ..............................................239 Angular Velocity.......................................246 Applied Pressure .....................................284 Applied Sinusoidal Drive Voltage ............258 Arbitrary Interface34, 67, 108, 165, 178, 236

Arbitrary Interface Boundary Conditions..................................... 34, 67, 108, 178

Arrhenius .........................................128, 136 Assemble .................................................241 Avogadro’s Number.................................127 Axial Water Pump Results.......................225 Axisymmetric ...........................276, 278, 303

choose..................................................276 axisymmetric 2D ........................................37 B Backward Angle.......................................246 Backward Facing Step.......................54, 120

Laminar Flow Past .................................54 Turbulent Flow Past .......................54, 120

Binary Diffusion........................................159 Biomechanics ............................................54 Bird-Carreau ..............................................25 Blackbody ................................................186

Spectral Emissive Power .....................186 Body Forces.............................................241 body-fitted-coordinate..............................194 Boiling/Condensating Flows ....................228 Boltzmann................................................163 Boltzmann’s .............................................186 Bordelon ..........................................231, 239 Boundary Condition Subtypes...................37 Boundary Conditions ...............................241 Boundary-Layer Flows.............................120 Boussinesq ................................................86

Boussinesq’s........................................... 102 Brick Element Option .............................. 269 Bubble Dynamics.................................... 239 Bump......................................................... 54 Buoyant Flows ........................................ 228 C Calculator Tool........................................ 311 Cantilevered Beam ................................. 312

Large Deflection .................................. 312 Carreau ..................................................... 16 Carreau Law ............................................. 25 Cartesian Strain Tensor.......................... 300 Cartesian Stress Tensor ................. 300, 303 Casson Model ........................................... 25 Cavitating Flows...................................... 228 Cavitation .. 59, 154, 223, 232, 233, 234, 239 Cavitation Model ............................. 235, 239 Cavitation Model Options........................ 235 Cavitation Module .... 25, 222, 223, 225, 227,

228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239

Cavitation Module Theory....................... 229 Cavitation Zone....................................... 239

Nuclei Population Downstream........... 239 Centrifugal Pumps .................................. 225 CGS ........................................................ 293 CGS Solver ............................................. 293 CGS+Pre......................... 115, 168, 182, 237 Chemistry Applications ........................... 122 Chemistry Module...... 25, 61, 108, 122, 123,

124, 127, 154, 155, 162, 165, 166, 167, 168, 169, 170, 171, 173

Chemistry Module Theory....................... 127 Chen ....................................................... 102 circumferential................................. 311, 314 Combustion............................................. 173 Combustion Reaction Equilibrium........... 173 combustors ............................................. 122 compared ................................................ 241

TFI ....................................................... 241 Complex Flows Including Separation ..... 120

Near-Wall Turbulence Models............. 120 compressibility ........................................ 231 Computational Fluid Dynamics . 77, 184, 228 Computer Methods ................................. 316 Concentration.......................................... 159 conduction/convection ...................... 56, 215 conduction+convection+radiation ........... 214 Conductivity Calculation ........................... 61 Conductivity Evaluation Methods.............. 61 Conjugate Gradient Solver ..................... 241 Conjugate Heat Transfer .......................... 56

321


Contact Analysis ......................................289 Contact Gap.............................................289 Contact Target Pair dialog.......................289 Contact Target Pair window ....................289 contact/target ...........................................289 Convection.......................................177, 221

Complex Geometries ...........................221 convection-diffusion-reaction...................152

governing .............................................152 convection-driven.....................................132 Coupled Flow...........................................303 Coupled Fluid/Thermal/Structural Analysis

.............................................................257 Orifice...................................................257

Coupled Radiation ...................................221 Numerical Simulation ...........................221

Coupled Solver ........................................126 Coupling Frequency ................................292 Critical Damping ..............................265, 276 Critical Pressure Scaling .........................239

Schiebe Headform Traveling Bubble Cavitation Inception..........................239

cross-diffusion............................................86 portion ....................................................86 value.......................................................86

Current Point Load ..................................285 Current Scalar..........................177, 178, 180 Cyclic Boundary Conditions.....................290 cyclic/periodic BC ....................................290 D Damping ..........................................276, 316 Damping Applications..............................316

Vibration Control ..................................316 Data Emissivity Sets ................................207 Database Manager ..................124, 159, 163 decoupled ................................................214 Deformation .............................................243 Deformation Module ........................223, 269

activated...............................................269 Deforming Grids.......................................269 Density.......................................25, 282, 300

Volume Condition Inputs........................25 Deposition Rate .......................................170 Desired Effect ..........................................178 determine.................................102, 212, 241

emissivity..............................................212 Development............................................120

Turbulence Models ..............................120 Devices ....................................................239 Dielectric Matrix .......................................282 differencing ......... 41, 72, 115, 168, 182, 237 Different Temperatures............................186 Different Types ........................................174

Transport Mechanisms ........................174 DIFFUSE Reflection Characteristic .........216 Diffusion...................................................177

diffusivities ...................... 133, 137, 141, 153 diffusivity 108, 124, 132, 150, 163, 174, 175,

177, 182 Direct Numerical Simulations.................... 91 Direct Solver ........................................... 293 Direction.................................................. 102 Dirichlet ................................................... 178 Discrete Ordinate Method...... 185, 186, 192,

195, 199, 202, 203, 206, 210, 211, 212 Discrete Ordinates Method. .................... 221 discrete-ordinate ..................... 192, 193, 194

solving ................................................. 194 discretized....................................... 193, 194 Displacement Field Contours.................. 258

High Frequency Resonator ................. 258 Dissipation Rate...................................... 119 distance................................................... 105

adjacent-to-wall ................................... 105 divergence/non-convergence ................. 303 DNS .......................................................... 91 DOM....... 185, 202, 203, 206, 207, 208, 209,

210, 211, 212, 213 Absorption Coefficient Settings ........... 210 Radiation Model Widow-Absorption

Coefficient Settings.......................... 211 Radiation Model Window-Emissivity

Settings............................................ 208 DOM Method........................................... 203

Radiation Model Settings .................... 203 Domain............................................ 102, 241 Domain Definition.................................... 248 Domain Division...................................... 248 domains containing................................. 241 DRey ....................................................... 102 DTF ......................................................... 297 Dynamic Coefficient ................................ 118 Dynamic Viscosity..................................... 25 Dynamics ........................................ 107, 239 E Eddy Viscosity.. 79, 102, 109, 114, 117, 118,

119 Effective Viscosity ............................. 43, 117 Eigenvalue .............................................. 303 elastic/elastic........................................... 262 elastic/rigid .............................................. 262 elastoplastic .................................... 269, 303 Electric ............................................ 256, 284 Electric Module ............................... 258, 284 electrochemistry.............................. 122, 155 electron-induced ..................................... 125 Electrophysics..................................... 15, 56 Electrostatic Field Distribution ................ 258 Electrostatic Forces ................................ 303 Electrostatic Load ................................... 258

Accelerometer Under .......................... 258 Element Conversion................................ 269

322

Index

Element Order .........................................269 Set........................................................269

Element Order menu ...............................276 Element Types.........................................269 Elements..........................................102, 241 Ellipsometry .............................................221 Elsevier Noth-Holland..............................221 emissivities ..............................................200

specifying .............................................200 emissivity . 67, 188, 189, 195, 197, 202, 203,

204, 206, 207, 208, 212, 213, 216 determine .............................................212 number .........................................207, 208 Patch ....................................................195

Emissivity Set ..........................207, 208, 212 emittance .................................................189

terms ....................................................189 Emitting-Absorbing ..................................189 employing ........................................102, 124

Boussinesq’s ........................................102 Stefan-Maxwell ....................................124

Energy Combust ......................................120 Engineering Flows ...................................228 Engineering Science................................316 Enhanced Brick........................................269 Enhanced Brick Elements .......................303 Environment Temperature.......................204 EOA Growth Process ..............................148 Equilibrium...............................................155 Equilibrium Reaction Model.....................125 Equilibrium Reactions..............................155 Equivalent Circuit Model............................54

Squeeze Gas Film .................................54 Exit Boundary ..........................................193 Expansion Coefficient......................278, 282 experimentation ...............................230, 235 Expression .......................................283, 284

choose..................................................283 Extension .................................................120

k-w-SST ...............................................120 External Heat Transfer ..............................67 Extrapolated...............................................37 Extremely High ........................................204 F farfield ................................................37, 214 farfield BC................................................214 Fast Monte Carlo Scheme.......................221

Thermal Radiation................................221 Favre-averaged .......................................141 Favre-averaged PDF ...............................141 Feature Scale Coupling ...........................170 Features-Automatic Re-meshing.............241 Features-User Defined Remeshing.........243 fi 241 Figure.......................................................102 Filtering Approach....................................120

Final Grid After Dividing Domain ............ 248 Final Grid After Motion............................ 248 Finite Element ................................. 241, 303 Finite Element Analysis .......................... 316

Applications ......................................... 316 Finite Element Solution........................... 241 Finite Element Stress.......................... 15, 56 Finite-Rate .............................................. 155 Finite-Rate Model.................................... 125

Species Solution.................................. 125 First Order............................... 269, 295, 303 Fixed Flux ............................................... 178 Fixed Mass Flow Rate .............................. 34 Fixed Mixture .......................................... 166 Fixed Pressure.................................... 34, 37 Fixed Total Pressure................................. 34 Fixed Value ............................................. 178 Fixed Velocity...................................... 34, 37 Fluid Effects ............................................ 303 Fluid Machinery....................................... 239 Fluid Mechanics 238............................... 120 Fluid Properties....................................... 232 Fluid Volumes ......................................... 174 fluid/solid ......................................... 175, 303 fluid/thermal/structural ............................ 257 Flux ........................................... 75, 166, 195 Flux Option................................................ 67 Forward Angle......................................... 246

specifying ............................................ 246 Free Simulations..................................... 120

Turbulent Reactive Flows.................... 120 Free Surface ............. 15, 154, 232, 233, 255 Free Surface Module ................................ 59 Free, Prescribed Displacement .............. 282 Frequency ............................................... 265

Raleigh Damping................................. 265 Frequency Iterations ............................... 292

Coupling .............................................. 292 Frequently Asked Questions.... 45, 119, 172,

184, 216, 239, 255, 303 FSI .................................................. 292, 303 FSI-related .............................................. 303 Full Algorithm.......................................... 161 Full Cavitation Model .............................. 239

Application........................................... 239 Full Cavitation Model. ............................. 239 fully-turbulent .......................................... 102 fully-turbulent region ............................... 102 Fung.................................................... 25, 54 G gain/loss.................................................. 194 Gas Diffusion .......................................... 239

The Effect ............................................ 239 Gas Liquid Flow ...................................... 239 Gas Phase ...................................... 155, 161

displays ............................................... 155

323


Gas Phase Reactions..............................123 Gas Turbine Atomizer..............................256

Thermoelastic Analysis ........................256 Gas Turbine Combustor ..............54, 77, 173 Gases ......................................................189

Theory-Radiation Characteristics.........189 Gaussian..........................................111, 302

specify ..................................................111 Gaussian Random Option .......................111 General Circulation Experiments.............120 General Liquid Chemistry ........................159 General Liquid Chemistry Option ............159 General Rate ...........................................125 General Scalar.................................174, 177

create ...................................................177 Generic ....................................................170 Generic Semiconductor Reactor .........54, 77 Geometric Non-Linear .............................275 geometrical ..............................................197 Geometry .................................................309 geometry/grid...........................................256 geometry/material ....................................282 Germano............................................91, 120 Germano Subgrid Scale Closure Method120

Proposed Modification .........................120 gerotor .....................................................223 Gibbs .......................................125, 128, 134

minimizing ....................................125, 134 Giersiepen .................................................54 Givi...........................................................120 governing .................................................152

convection-diffusion-reaction ...............152 gradients/scales.......................................216 Graphic ................... 117, 170, 182, 215, 238 Graphic Tab.............................................300 Graphical Output..... 115, 182, 215, 238, 302 GRAY.......................................................216 GRAY Gray/Non-gray..............................216 Grid Deformation Applications.................240 Grid Deformation Boundary Conditions...246 Grid Deformation Features ......................241 Grid Deformation Implementation............244 Grid Deformation Module240, 241, 243, 244,

245, 246, 248, 255, 273, 294 activated...............................................294 Describes .............................................244

Grid Deformation Module Mode ..............245 Grid Motion ..............................................294

checking ...............................................294 Grid Motion Option Available...................294 Grid Parameter ........................................105 gridding ...... 22, 59, 105, 154, 176, 200, 232,

244, 269 H Handbook ................................................221

Optical Constants.................................221

Hardening ............................................... 278 Parameters N/m2 ................................ 278

Heart Valve Prostheses ............................ 54 Heat Flux................................................... 67 Heat Module.............................................. 75 Heat Subtypes .......................................... 67 Heat Transfer ..... 59, 72, 120, 173, 221, 233,

256, 257 k-e Predictions..................................... 120

Heat Transfer Implementation .................. 59 Heat Transfer Module 25, 40, 56, 57, 58, 59,

60, 61, 67, 72, 75, 77, 108, 201, 232, 257, 258, 277 affect...................................................... 60 CFD-ACE-Solver ................................... 59 If 40 use................................................... 56, 77 values .................................................... 72

Heat Transfer Module Output ................... 74 Heat Transfer Settings Mode.................... 60 heatflux ................................................... 196 Heating...................................................... 67 Helium..................................................... 235 Hellsten ................................................... 120 hematocrit ................................................. 25 hemolysis .................................................. 16

stress..................................................... 16 Hemolysis Model....................................... 16 hexahedrals ............................ 268, 269, 276 hf 67 HI ............................................................ 239 High......................................................... 204 High Frequency Resonator..................... 258

Displacement Field Contours .............. 258 High Order Wall Local......................... 24, 39 High Reynolds..................................... 79, 83 Highly Non-linear Geometry ................... 248 high-Re-number ........................................ 79 high-Reynolds ........................................... 84 high-Reynolds-number ............................. 79 Hilton Head ............................................. 239 Hinze............................................... 231, 239 HO2......................................................... 128 Honolulu.................................................. 239 Hoop Stress ............................................ 311 Howell ..................................... 186, 191, 221 Hydraulic Diameter ................................. 111 Hydrofoil.................................................. 239 Hydrogen-Air Combustion ...................... 173 hyper-ellipsoid......................................... 148

estimate............................................... 148 I ic1 ............................................................. 67 ic2 ............................................................. 67 Ice Melting..................................... 57, 60, 66

activated ................................................ 66

324

Index

Ideal Gas Law............................................25 If 241

Heat Transfer Module ............................40 ill-conditioned...........................................293 Im.............................................................193 Image.......................................................257 Implicit Damping ......................................276 Implicit Motion..........................................255 Implicit Pressure ......................................284 Implicit Shear Stress................................285 Improved Wall Treatment. .......................120 In 102, 241

Rayleigh Damping................................276 In Critical Damping ..................................276 In Situ Adaptive Tabulation..............126, 148 in V2006...................................................102 in/outs ......................................................111 Inc ..............................................................77 includes....................................................105

k-w........................................................105 incoming ..................................................189

W‘ .........................................................189 increase ...........................................256, 277

100K.....................................................277 500K.....................................................256

Inducers ...................................................239 inert/passive.............................................174 Influence ..................................................239

Investigations Concerning....................239 Initial Conditions 40, 72, 114, 148, 167, 180,

215, 236, 291 specify ..................................................167

Initial Conditions Panel ...... 40, 72, 114, 167, 180, 291 see .........................................40, 180, 291

Initial Temperature...................................291 Initial Yield Stress ....................................278 Inlet ............. 34, 67, 108, 165, 178, 213, 246 In-plane Constraint ..................................285 Instantaneous ..........................................155 Instantaneous Chemistry Model ..............153 Instantaneous Reaction Model ................125 Instantaneous Reactions .........................155 Integrated Mean Reaction LEM...............161 integro-differential ....................................189 Interface...................................................178 introduced—a ..........................................276 Investigations Concerning .......................239

Influence...............................................239 invoking....................................................195

McMahon .............................................195 involving...................................126, 128, 196

1000 .....................................................196 Nsp.......................................................128 surface-adsorbed .................................126

Inward ......................................................214

set........................................................ 214 ION.......................................................... 170 ioper ........................................................ 170 Irrelevant ................................................. 216

ACEU .................................................. 216 ISAT ................................ 126, 147, 148, 161 ISAT Threshold ....................................... 161 ISAT Tolerance....................................... 161 Isothermal ........................................... 66, 67 Isothermal Assumption ........................... 232 Isothermal Option...................................... 67 isotropy-of-the-small-scales...................... 91 ISROMAC-8 ............................................ 239 it’s.................................................... 211, 214 iter ........................................................... 292 Iteration Tab............................................ 292 Iterations ................................................. 292 Iterative Methods .................................... 316

Sparse Linear Systems ....................... 316 ith .................................... 127, 128, 132, 241 J J.C........................................................... 316 J.O .......................................................... 239 J.R........................................................... 221 J/kg-K........................................................ 75 J/s ........................................................... 215 jth ............................................................ 241 K Kato-Launder k-e ...................................... 83 k-e ........................................................... 102 Kelvin ...................................................... 170 Kinematic Viscosity ................................... 25 Kinetic Theory ......................................... 155 Kirchhoff’s ............................................... 188 Kolmogorov............................................. 150

applying ............................................... 150 Kronecker................................................ 241 L Lagrangian ........................................ 16, 134 Lame ....................................................... 241 Laminar ........................................... 102, 173 Laminar Chemistry Operator Splitting..... 126 Laminar Viscosity.............................. 43, 102 Large Eddy Simulations..... 78, 91, 105, 107,

115, 126, 147, 152, 161 Launder................................................... 102 Layer Model ............................................ 102 LES ..... 78, 91, 105, 107, 114, 115, 118, 152 LES Module Graphical Output ................ 118 Li 128, 134, 239

respect................................................. 134 Lilly .......................................................... 120 Limitations-Arbitrary Interfaces ............... 268 Limitations-Cyclic/Thin Wall Boundary

Conditions ........................................... 268 Limitations-Limited Element Library........ 268

325


Limits ...................... 116, 169, 180, 182, 237 LINEAR....................................................282 Linear Analysis ........................................275 Linear Anisotropic Materials ....................278

choose..................................................278 Material Property Inputs.......................278

Linear Eddy Model...........................126, 173 Turbulent Scalar Transport ..................173

Linear Eddy Sub-Grid Model ...................173 Turbulent Reacting Flows ....................173

Linear Extrapolation.................................161 Linear Isotropic Materials ........................278

Material Property Inputs.......................278 Linear Lateral Resonator Comb Drive.....258 lines .................................................170, 216

EVOLVE...............................................170 Liq ..............................................................25 Liquid Phase............................................159 Liquid Water.............................................197 liquid’s......................................................231 Liquids .............................................166, 234 lists...........................................................216

n0 .........................................................216 lj 134 Lm....................................................102, 189 Load Subtype...........................................284 Loads .......................................282, 283, 285 Local ........................................................102 Local Material ..........................................278 Local X-Axis Vectors ...............................278 Local Y-Axis Vectors ...............................278 Localized Dynamic ..................107, 118, 161 Localized Dynamic Subgrid Scale Model 120

Turbulent Wall-Bounded Flows............120 Los Alamos National Laboratory Technical

Report LA-UR-97-03892 ......................316 Low ..........................................................204 Low Reynolds Number Turbulence Model

.............................................................120 low-Reynolds ...............................83, 84, 105

form ........................................................83 low-Reynolds-number................................79 LPCVD2...................................................170 Lumped - In..............................................276 M M.E ..........................................................316 M.M..........................................................239 M.W .........................................................120 m/m-K ..............................................278, 282 m/s .............................................43, 118, 257 m/Sc.........................................................177 m’ .............................................................193 m0..............................................................25 m-1...........................................................216 m2......................................................67, 195 m2/s ................ 117, 118, 119, 170, 177, 182

m2/s2 ................................ 75, 117, 118, 119 m2/s3 ...................................... 117, 118, 119 m-3.......................................................... 127 m3-sec .................................................... 171 Mach ......................................................... 37 Mach Number ........................................... 43 Malasekera ....................................... 77, 184 Malkus............................................. 265, 316 March 2000 ............................................. 239 Mass Conservation ................................... 17 Mass Diffusion ........................................ 162 Mass Flow Calculations ............................ 15 Mass Fraction ......................................... 125 Mass Matrix Type menu ......................... 276 Mass Proportional Damping ................... 265

Total Damping Included ...................... 265 MassFr .................................................... 238 Material Non-Linear ................................ 275 Material Property Inputs.......................... 278

Linear Anisotropic Materials................ 278 Linear Isotropic Materials .................... 278 Non-Linear Isotropic Materials ............ 278

Material Type .......................................... 278 Material Type menu ........................ 278, 282 Materials ......................... 216, 241, 278, 303

Number................................................ 216 Mathematical Basis................................. 239 Max ......................................................... 292 Max dphi Energy Norm ........................... 303 Maximum Additions................................. 161 Maximum Plot Displacement .................. 297

opens................................................... 297 Maximum Temperature........................... 161

setting.................................................. 161 Mazunder ........................................ 196, 221 MC .......................... 196, 202, 212, 214, 216

performing ................................... 196, 202 McGraw Hill............................................. 221 McGraw-Hill ............................................ 316 McMahon ................................................ 195

invoking ............................................... 195 McMurtry ................................. 150, 152, 173 measure .................................................... 86

vorticity .................................................. 86 Mech ....................................................... 120 Mechanical Engineering ......................... 173

Thermal Energy Laboratory Department......................................................... 173

Media ...................................................... 155 MEMs.............................................. 248, 258 Menon ............................................. 120, 173 Menter............................................... 86, 120 mesopump .............................................. 258

shows .................................................. 258 Mesopump Cell ....................................... 258

Solid Model.......................................... 258

326

Index

meters.eg.................................................216 Method.....................................241, 243, 316

Computation.........................................316 method uses ............................................241 mI.....................................................127, 134 min/max ...................................................237 Mindle ......................................................316 Minf ............................................................25 minimizing........................................125, 134

Gibbs............................................125, 134 Minimum Temperature ............................161 Mix JANNAF1 ............................................61 Mix Kinetic Theory1 ...................................61 Mix Kinetic Theory2 ...................................25 Mix Polynomial.....................................25, 61 Mix Sutherland’s Law2 ..............................25 Mixer Channel Geometry.........................257 Mixer Channel Grid..................................257 Mixture Fraction.......................................124 Mixture Fraction Approach ......................155 Mixture Mass Fraction ... 124, 125, 133, 155,

162 Models..................................................155

Mixture Solution...............................125, 136 Mixtures ...................................................162 Mixtures button ........................................159 mo............................................................170 Modal Analysis.........................................275 Modal Analysis Model..............................297 Mode Shape Output ................................297

value.....................................................297 Model Definition.......................................309 Model Explorer.........................................282 Model Options 106, 154, 163, 177, 180, 201,

234, 244, 245, 269, 274, 282, 289, 292 Model Options Panel ..... 106, 154, 155, 177,

201, 234, 244, 274 see ...... 106, 154, 177, 201, 234, 244, 274

Model Options Settings ...........................235 Model Options Shared.............................274 Model Options-Shared.............................276 Model Options-Stress-Modal Analysis ....297 Model Output ...........................................297 Model Setup.....................................202, 273 Model.0001.DTF......................................297 Model.0002.DTF......................................297

shape ...................................................297 model.spc ................................................248 modelname.DTF.... 117, 170, 182, 215, 238,

300 modelname.e,a ........................................170 modelname.EVSPEC ..............................170 modelname.nn.FSC file ...........................170 modelname.nnn.FSC...............................170 modelname.out .. 43, 75, 170, 171, 182, 183,

215, 216

modelname.species_name.nn.EVFLX ... 170 modelname>.OPTIC ............................... 216 modelname>.out file ............................... 255 model-name>.out file .............................. 255 modelname>.PATCH.............. 200, 214, 216 modelname>.PATCH file ........................ 200 Models menu .......................................... 202 Moderate................................................. 204 Modest, M.F ............................................ 221 Module Graphical Output........................ 170 module’s...................................... 15, 56, 123 Modules Not Supported .......................... 232 Modules-Stress Module .......................... 256 modulus .......................... 241, 275, 278, 282 Molecular Wt ............................................. 25 Monitor .............................................. 74, 301 Monitor Point Definition........................... 301 Monitor Tab............................................. 301 Monte Carlo ... 185, 195, 196, 197, 199, 200,

202, 216 choose................................................. 202 use....................................................... 196

Monte Carlo Method ....................... 195, 199 Monte Carlo Model.................................. 196 Monte Carlo Radiation Mode .................. 202 Monte Carlo raytracing ................... 196, 202

result.................................................... 196 Monte-Carlo Methods ............................. 186 Monthly Weather Review 91................... 120 Motion ..................................... 241, 243, 255 Moving ................................................ 57, 60

Solid ...................................................... 60 Solids..................................................... 57

Moving Edge........................................... 248 Moving Edge ae...................................... 248 Moving Grid............................................. 245 Moving Grid checkbox ............................ 245

selecting .............................................. 245 Moving Grid Setup .................................. 248 Moving Solids................................ 57, 60, 66

activated ................................................ 66 moving/deforming ................................... 240 Mpa ......................................................... 314 mt ............................................................ 102 mTorr ........................................................ 16 Mu_0 ......................................................... 25 Mu_inf ....................................................... 25 Mu0 ........................................................... 25 Muinf ......................................................... 25 Multi Domain Structures ......................... 248 Multi-component ..................... 124, 127, 163 Multi-component Diffusion .............. 155, 163 multi-dimensional .................................... 222 multi-disciplinary ..................................... 256 Multi-Physics Applications .................. 15, 56 multiplicative ................................... 150, 316

327


Multi-step Reaction......................54, 77, 173 Mushy ........................................................66

choose....................................................66 N Natural Convection ..............................54, 77 Natural Convection Problems....................56 Near-Wall Turbulence Models .................120

Complex Flows Including Separation ..120 Negative Volumes ...................................303 Negative Volumes Encountered..............303 Nesman ...................................................239 New Grid Becomes Orthogonal...............248 New Numerical Method ...........................221

Radiation Heat Transfer.......................221 Newmark..........................................274, 316

uses......................................................274 Newmark Scheme ...................................276 Newton-Raphson.............................262, 292 ng_Vol......................................................303 ni 127, 134

respect .................................................134 Nitrogen ...................................................235 No Melting..................................................66 No Solid Cell Wall Boundary .....................67 No Solidification.........................................66 noded tetrahedrals...................................313 Nodes ..............................................241, 243 non-condensable ... 222, 228, 231, 235, 236,

239 level......................................................236 pick.......................................................235

Non-Condensable Gases ........................231 Theory-Effect........................................231

non-dimensional ........................80, 216, 312 define .....................................................80

None ........................................................163 non-equilibrium ........................................102 Non-Equilibrium Model ............................102 Nongray ...................................................216 Non-Gray .........................................203, 216 NONGRAY Gray/Nongray .......................216 Non-Gray Property Banding ....................206 Nongray/Semitransparent........................216 Nonhomogeneous Participating Media. ..221 Non-Linear Analysis ................................275 Non-Linear Isotropic Materials.................278

choose..................................................278 Material Property Inputs.......................278

non-Newtonian...........................................16 Non-Newtonian Viscosity Options .............16 non-oscillatory..........................................265 Normal .....................................................102 Normal Distance ......................................102 Normal Stress ..........................................302 Normal Translation Motion ......................255 Normalized Tip Deflections......................312

NOx................................................. 124, 171 Nuclei Population Downstream............... 239

Cavitation Zone ................................... 239 Numerical Control Settings ..................... 180

User Scalar Variables ......................... 180 Numerical Damping ................................ 276 Numerical Methods................................. 316 Numerical Simulation...................... 120, 221

Coupled Radiation............................... 221 Surface Pressure Fluctuations ............ 120

O Omega ................................................ 39, 40 one-dimensional.............................. 147, 150

sum...................................................... 147 OPAQUE................................................. 216 opaque/semi-transparent........................ 212 opaque/transparent................................. 212 Operator Splitting ............................ 147, 161 Optical Constants.................................... 221

Handbook ............................................ 221 Optical Database File ..................... 200, 216 OPTICAL.DAT ........................................ 216 Optional Variables..................................... 37 Options...................... 23, 102, 155, 159, 177 Options Panel .. 60, 106, 201, 202, 234, 245,

274 order-of-magnitude ................................. 204 Orifice...................................................... 257

Coupled Fluid/Thermal/Structural Analysis......................................................... 257

Orszag .............................................. 82, 120 OUT .......................................................... 74 Outlet .......... 34, 67, 108, 165, 178, 213, 246 Output 42, 74, 117, 170, 182, 215, 238, 296,

297, 299, 300 Output Elapsed Time .............................. 303

End ...................................................... 303 Output File Snippet ................................. 303 Output Frequency ................................... 297 Output Location menu ............................ 297

contains ............................................... 297 Output Panel ..... 74, 296, 297, 299, 300, 301 Outward .................................................. 214

set........................................................ 214 Overview ................................................. 106 Oxford University Press .......................... 239 P P_tot.......................................................... 43 Pair Name............................................... 289 Palik ................................................ 197, 221 Panel's Graphic....................................... 300 Panel's Output ................................ 297, 299 parameterizations ................................... 152 parameterized ......................................... 152 Parameters N/m2.................................... 278

Hardening............................................ 278

328

Index

Parametric ...............................................285 Parametric Input dialog............................285

clear .....................................................285 Parametric Input window .........................285 PARTIALLY_SPECULAR........................216 Passive ............................................174, 177 Passive Scalar .................................174, 177

create ...................................................177 Patch........................................................195

Emissivity .............................................195 Patch File.........................................200, 216 Patel.........................................................102 pdf.. 124, 125, 126, 141, 142, 143, 144, 147,

150 form ......................................................141 shape ...................................................126

phenomenologically...................................79 Phys.................................................278, 282 Phys.Fluids ..............................................120 Physical ...................................................102 pI 283, 284 Piecewise Linear..................................25, 61 Piezo........................................................282 Piezoelectric Coupling Matrix ..................282 Piezoelectric Properties...........................282 Planck’s ...................................................186 Planck's....................................................186 Plane Strain .....................................276, 303 Plane Stress ....................................276, 303 Plesha......................................................316 Point Load................................................285 point-iterative ...........................................137 Poiseuille ...................................................16 Poisson ................... 174, 182, 269, 278, 303 Poisson Scalar.................................174, 177

create ...................................................177 Poisson’s .................................................241 Poisson's Ratio ................................278, 282 Polarized Light .........................................221 Polynomial Constants................................61 Porous Media...........................................101 Post Processing Variables. 43, 75, 171, 183,

302 Post-Processing Variables ......................238 Power Law.................................................25 ppm..........................................................231 Prandtl ...................................58, 61, 80, 108 Prandtl Number..........................................61 Pratt .........................................................125 Pratt David ...............................................173 Prediction.................................................120

Channel................................................120 pre-exponential ........................................128 Prentice-Hall ............................................316 Prescribed Displacement.........................283 Pressure ......................... 102, 269, 309, 311

velocity sensitized ............................... 102 Pressure Distribution .............................. 239 Pressure Field Calculations ...................... 15 Pressure Gradient................................... 102 Pressure Profiles Without Cavitation Model

............................................................ 223 Previous .................................................. 115 Primitive Equations ................................. 120 Principal Strain........................................ 300 Principal Stress ....................................... 300 Principal Stress Direction........................ 300 Printed Output......................................... 215 Prisms ..................................................... 241 Problem Type..... 23, 59, 106, 154, 177, 201,

233, 244, 273 Problem Type Panel .. 23, 59, 106, 154, 177,

201, 233, 244, 273 Problem Type/Modules........................... 309 problem-specific...................................... 303 Profile...................................................... 102 Profile 2D .......................................... 67, 110 Progress Variable ................................... 171 Propane .............................. 54, 77, 120, 173

Turbulent Mixing.............. 54, 77, 120, 173 Properties... 25, 61, 162, 177, 216, 235, 241,

278, 282 Name................................................... 216 set.................... 25, 61, 162, 177, 235, 278

Property Database Manager................... 167 Property Manager ................................... 165 proportionality ................................... 79, 133 Proposed Modification ............................ 120

Germano Subgrid Scale Closure Method......................................................... 120

Przekwas ................................................ 221 Pseudo MC ............................................. 202 Pump............................................... 239, 258 Pure Conduction Problems....................... 56 Pyramidal ................................................ 303 Pyramids ................................................. 241 Q quadrature............................................... 191 Quasi MC................................................ 202 R Radial Displacement Contours ............... 311 Radiation................. 195, 199, 201, 213, 233 Radiation Heat Transfer.......................... 221

New Numerical Method....................... 221 Radiation Mode....................................... 201 Radiation Model ...................... 202, 214, 215 Radiation Model Settings........................ 203

DOM Method....................................... 203 Radiation Model Widow-Absorption

Coefficient Settings ............................. 211 DOM.................................................... 211

Radiation Model window......... 202, 203, 204

329


STS Method .........................................204 Radiation Model Window-Emissivity Settings

.............................................................208 DOM.....................................................208

Radiation Model Window-Radiation Sources Tab .......................................................211

Radiation Model-Absorption Coefficient window .........................................210, 211

Radiation Module..... 56, 185, 186, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 199, 200, 201, 202, 203, 204, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 221 affect ............................................201, 215 Describes .............................................200 respect .................................................213 tell.........................................................216

Radiation Module Related Graphical Output.............................................................215

Radiation Source .....................202, 204, 211 press ....................................................211

Radiation Temperature............................214 Radiation Wavelengths Settings..............206 Radiation Window-Emissivity Settings ....207

STS ......................................................207 Radiative Heat Transfer...................189, 221 Radiative Wall Heat Flux .........................215 radiatively.................................................195 radiosity ...................................................191 Raleigh Damping .............................265, 276

Frequency ............................................265 Ramp .........................................................54 Random Inlet ...........................................111 RANS............................ 78, 79, 91, 105, 114 RANS checkbox.......................................114 Rapid Thermal Chemical Vapor Deposition

.....................................................185, 195 Rapid Thermal Processing ..............185, 195

realized.................................................195 rate/Specific .............................................110 rate-of-progress .......................................128 Rayleigh...................................................265 Rayleigh Damping ...................................276

In 276 Rayleigh-Plesset......................................230

form ......................................................230 raytracing .................................................195 Re*y .........................................................102 Reacting...................................................228

Flows....................................................228 Reaction Cutoff Temperature ..................161

set ........................................................161 Reaction Database Manager...................155 Reaction Forces...............................299, 300 Reaction Manager ...................................124 Reaction Rates ................................170, 171

realized ................................................... 195 Rapid Thermal Processing.................. 195

recirculation............................................. 119 RNG .................................................... 119

Reduced Flow Models ........................ 16, 39 reduces ................................................... 241 Reference ....................................... 163, 230

provide................................................. 163 Reference Specie ........................... 124, 163 reflectivities ............................................. 200 Region..................................... 102, 241, 243 Regular Translation ................................ 255 Reisman.......................................... 231, 239 relaminarization......................................... 83 Relations ................................................. 127 Relations-Chemical Rate Expressions ... 128 Relations-Composition Variables............ 127 Relations-Introduction............................. 127 Relations-Mixture Fractions .................... 132 Relax Tab................................................ 294 Relaxation ............................... 115, 168, 237 releases .................................................. 232

CFD-ACE ............................................ 232 remesh ............................................ 241, 245 remeshed ................................................ 244 remeshing ............................... 241, 243, 244 renormalization ......................................... 82 Renormalization Group Analysis ............ 120 Replace Simulation................................. 297 Required Inputs................................... 25, 61 Required Variables ................................... 37 Reset button............................................ 285

use....................................................... 285 Results Table .......................................... 314 Rey.......................................................... 102 Rey Re*y................................................. 102 Reynolds Averaged Navier-Stokes........... 78 Reynolds stress ...................................... 102 Reynolds-number...................................... 79 RMS ................................................ 111, 114 RNG .................................................. 82, 119

recirculation ......................................... 119 RNG k-e .................................................... 82 root-mean-square ................................... 110 Rotating................................................... 178

Walls.................................................... 178 rotating/deforming................................... 223 Rotation................................................... 255 Rotation Angle ........................................ 246 Rotation Input Variables ......................... 246 Rotation Machinery. ................................ 239 Rotation Omega...................................... 246 RPM ........................................................ 223 S Sample Grid ............................................ 269

Analysis ............................................... 269

330

Index

Sandaram ................................................105 Save LES Statistics .................................115 SC Panel..................................................293 Scalar.......................................177, 178, 180 Scalar Control ..........................................174 Scalar Diffusion Coefficient .....................183 Scalar Diffusivity ..............................177, 182

Volume Condition Inputs......................177 Scalar Module..........................................183 Scalar Module Graphical Output .............182 Scalar Name....................................177, 183 Scalar Tab ...............................................177 Scalar Types............................................174 Scalar Values...........................................182 scalar’s diffusivity.....................................177 Scale Temperature ..................................161 scales...............................................102, 111 scales lt....................................................102 Scaling Factor..........................................297

use .......................................................297 Scattering Gray Medium..........................189 scheme ............................................241, 243 scheme applies........................................241 Schiebe Headform Traveling Bubble

Cavitation Inception .............................239 Critical Pressure Scaling......................239

Schmidt........................... 108, 124, 150, 177 Schmidt Number..............................162, 177 Schneck .....................................................25 Schuller....................................................239 Second Order ..........................269, 295, 303 Semi.........................................................216 semi-independence .................................303 semi-transparent............. 199, 209, 210, 211 Set Residual Frequency ..........................299 Setup ...............................................154, 176 Setup-Problem Type..........................23, 273 SGS ...........................................91, 107, 109 SGS Dissipation Rate..............................118 SGS Kinetic Energy.................................118 SGS Model ..............................................107 SGS Turbulent Kinetic Energy.................109 Shared . 23, 60, 61, 155, 177, 201, 234, 245,

292 Shared - When.........................................295 Shared Sections ......................................301 Shared Tab..............................................177 Shear Flows.............................................120 Shear Modulus.........................................278 Shear Stress ............................................102 Shear Stress-Related Blood Damage .......54

Estimation ..............................................54 Shearmax ................................................302 ShearMin .................................................302 Shell Problems.........................................316 Shell Surface ...................................269, 303

Silicon Dioxide ........................................ 197 Simple Flow Model checkbox ................... 24

Selecting................................................ 24 simplifies ................................................. 241 Single-Step Finite Rate Chemistry Model153 Singular Value Decomposition................ 161 Situ Adaptive Tabulation......................... 147 Slip Wall Boundary Conditions ................. 16 Slower ..................................................... 116 Smagorinsky ............................. 91, 107, 120 Smagorinsky SGS................................... 107 Solid ...... 57, 60, 61, 177, 221, 277, 278, 282

Moving............................................. 57, 60 set........................................................ 278

Solid Cell................................................... 67 Solid Model ............................................. 258

Mesopump Cell ................................... 258 Solid Shells ..................................... 269, 303 Solid VC Type......................................... 282 Solid Volumes......................................... 174 solid/fluid ................................................. 268 solid/solid ................................................ 175 Solid-body Elasticity Analogy.................. 241 Solidification.................................. 57, 60, 66

activated ................................................ 66 Solve Combustion................................... 161 Solve Concentration ............................... 159 Solvers Tab............................................. 293 Some Preconditioning Techniques......... 316

An Assessment ................................... 316 Soret ....................................................... 163 source/sink................................................ 57 Spalart Allmaras Model........................... 114 Spalart-Allmaras ............................... 88, 119 Spalart-Allmaras Model .......................... 109 Spalding .................................................... 80 Spalding’s ............................................... 102 Sparse Linear Systems........................... 316

Iterative Methods................................. 316 Spatial Differencing.... 41, 72, 115, 168, 180,

182, 237 Spaulding .......................................... 80, 120 SPC......................................................... 248 SPC file ........................................... 248, 303 Species button ........................................ 159 Species Conservation Options ............... 124 Species Diffusivity................................... 170 Species Flux ........................................... 170 Species Fraction Approach..................... 155 Species Mass.......................................... 171 Species Mass Fraction .. 124, 125, 133, 155,

161, 162, 170 use....................................................... 170

Species Mass Fraction Mass Transport . 155 Species Solution ............................. 125, 137

Finite-Rate Model ................................ 125

331


Species Specification ..............................166 Species Thermodiffusivity........................170 Specific Heat........................................61, 75 Specific Heat Calculation...........................61 Specific Heat Evaluation Methods.............61 Specify Contact Target button .................289 Spectral Characteristic ............................216 Spectral Emissive Power.........................186

Blackbody.............................................186 specular .................. 185, 193, 199, 200, 216 SPECULAR Reflection Characteristic .....216 specularity................................................216 SPEEDIE .................................................170

corresponding ......................................170 SPEEDIE DEPO......................................170

correspond ...........................................170 SPEEDIE LPCVD ....................................170 SPEEDIE LPCVD1 ..................................170 Speziale ...................................................120 Spray .......................................................233 Spring Force ............................................285 Spring Force Option ................................285 Springer .....................................................54 Squeeze Gas Film .....................................54

Equivalent Circuit Model ........................54 Srinivasan ................................................221 ST Opaque/Semitransparent...................216 Staggered Chemistry.......................126, 161

and/or ...................................................126 Standard ..........................................102, 241 Standard Brick Elements .........................303 standard Galerkin ....................................241 Static Pressure ..........................................43 Stefan Maxwell ........................................163

form ......................................................163 Stefan-Boltzmann ............. 67, 186, 191, 195 Stefan-Maxwell ................................124, 163

employing.............................................124 request .................................................163

Stiffness ...........................................241, 265 Stoffel...............................................231, 239 stoichastically...........................................150 stoichiometric.......... 125, 127, 128, 133, 136

array .....................................................128 stoichiometrically .....................125, 133, 134

specify ..........................................125, 134 stoichiometry............................125, 127, 136 STRAIN......................................43, 313, 316 Strain Invariant.........................................118 strain-displacement .................................262 Stream Function ........................................43 Stress16, 102, 241, 255, 256, 258, 269, 273,

274, 277, 282, 289, 290, 292, 293, 295, 299, 300, 303, 309, 313, 316 Formulas ......................................313, 316 hemolysis ...............................................16

solving ................................................. 255 VOF ..................................................... 255

Stress - Damping Responses ................. 265 Stress Analysis ....................................... 311 Stress Applications ................................. 256 Stress Boundary Conditions ................... 282 Stress Calculation................................... 309 Stress Concentration .............................. 313 Stress Concentration Geometry ............. 309 Stress Concentration Grid ...................... 309 Stress Equation Options ......................... 293 Stress Mode............................................ 277 Stress Model Options ............................. 274 Stress Parameters .................................. 292 Stress Solver Control .............................. 292 Stress Tab....................................... 269, 274 Stress Tab Analysis ................................ 275 Stress/Strain ........................................... 303 Structural Analysis .................................. 269 Structural Analysis Module ..................... 303 Structural Damping ................................. 276 Structure ......................................... 241, 243 STS 204, 207, 209, 210, 211, 212, 213, 214,

216 appear ................................................. 207 Radiation Window-Emissivity Settings 207

STS Method .................................... 204, 207 Radiation Model Window .................... 204

subgrid ...................................... 91, 126, 150 mixing .................................................. 150

Sub-grid Linear Eddy Model ................... 161 Subgrid Scale.......................................... 107 Sub-iteration............................................ 204 sublayer ............ 79, 80, 83, 84, 85, 102, 105 sublayers................................................... 79 Substrate Material Name ........................ 216 SubType.................................................. 289 SubTypes.................................................. 37 sum ................................................. 117, 147

one-dimensional .................................. 147 turbulent .............................................. 117

SUMMATION.......................................... 166 Sundaram ............................................... 120 Supersonic Flows.............................. 54, 228 Surface Pressure Fluctuations................ 120

Numerical Simulation .......................... 120 Surface Property ..................................... 216 Surface Reaction 54, 77, 123, 155, 166, 173 Surface Reaction Manager ............. 126, 166

launch.................................................. 166 Surface Reflection Characteristics.......... 216 surface. ................................................... 120 surface-adsorbed.................................... 126

involving .............................................. 126 Surface-to-Surface. 185, 186, 199, 204, 212,

215, 216

332

Index

Sutherland’s Law .......................................25 Swirl Model ................................................16 Symmetry.................................178, 282, 288

choosing...............................................288 T Tangential Vectors...................................290 Target Surface Parameters .....................289 Target Surfaces .......................................289 TauMax....................................................302 tells ..........................................................216

Radiation Module .................................216 Temperature ......................................25, 163

function.................................................163 Temperature Intervals .............................161

number .................................................161 Temperature Limits....................................61 temperature-dependent ...........................128 term..................................................189, 241

emittance..............................................189 Test Filter Kinetic Energy ........................118 tet .................... 105, 154, 176, 200, 244, 269 Tetrahedral, Prismatic .............................303 tetrahedrals......................................241, 268 TFI ...........................................................241

compared .............................................241 TFI if.........................................................241 TFI scheme..............................................241 The key ....................................................102 Thermal....................................................303 Thermal Energy Laboratory Department.173

Mechanical Engineering.......................173 Thermal Field Calculations ........................56 Thermal Gap Model...................................72 Thermal Radiation ...................................221

Fast Monte Carlo Scheme ...................221 Thermal Radiation Heat Transfer ............221 Thermal Stresses.....................................314 thermo-diffusion.......................................163 Thermo-diffusion button...........................163

checking ...............................................163 thermoelastic .................. 256, 262, 277, 291

activated...............................................277 value.....................................................291

Thermoelastic Analysis....................256, 314 Gas Turbine Atomizer ..........................256

Thermoelasticity.......................................262 THigh .........................................................66 Thin-Wall Boundary Conditions .................34 Three Dimensional Radiative Heat-Transfer

.............................................................221 three-dimensional ....................................150 Time Accuracy .........................................274 Time Step Frequency ..............................297 Time Steps...............................................297

Ending ..................................................297 Starting.................................................297

Tools->Utilities ........................................ 309 use....................................................... 309

Total Damping Included.......................... 265 Mass Proportional Damping................ 265

Total Enthalpy ........................................... 75 Total Pairs............................................... 289 Total Point Loads.................................... 285 Total Pressure........................................... 43 Total Scalars ........................... 177, 178, 180 Total Temperature .................................... 75 Total Void Fraction.................................. 238 Total_Volume_Fraction........................... 238 Transfer Calculations................................ 56 Transfer Module... 56, 57, 58, 59, 60, 61, 66,

67, 71, 72, 74, 75, 77 transfinite ........................................ 241, 269

uses..................................................... 269 Transfinite Interpolation .................. 241, 248 Transient Computation ........................... 316 Transient Results .................................... 297 Translation .............................................. 255 transmissibility......................................... 188 Transonic Flows...................................... 228 Transparency Characteristics ................. 216 Transport Mechanisms ........................... 174

Different Types.................................... 174 Triangles, Quadrilaterals......................... 303 Triplet Map Illustration ............................ 150 turbo-machinery ...................................... 222 Turbomachinery ...................................... 225 turbo-pumps............................................ 222 Turbulence56, 102, 106, 108, 114, 120, 231,

233 Theory-Effect....................................... 231 use............................................... 120, 233

Turbulence Combustion Interaction........ 141 Turbulence Intensity ............................... 111 Turbulence Modeling .............................. 120

CFD ..................................................... 120 Development ....................................... 120

Turbulence Module .... 78, 79, 80, 82, 83, 84, 85, 86, 88, 91, 101, 102, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120 Describes ............................................ 105

Turbulence Module Related Graphical Output.................................................. 117

Turbulence Quantity ............................... 109 turbulence Reynolds ............................... 102 Turbulence Tab....................... 106, 107, 111 Turbulence Theory.................................... 78 turbulence/chemistry....................... 124, 125 turbulence-chemistry ...................... 150, 152

resolves ............................................... 152 turbulence-combustion ........................... 126

333


334

Turbulence-Combustion Interaction-Application............................................152 Large Eddy Simulation.........................152

Turbulence-Combustion Interaction-In Situ Adaptive Tabulation .............................148

Turbulence-Combustion Interaction-Operator Splitting .................................147

Turbulence-Combustion Interaction-Subgrid Linear Eddy Model ...............................150

Turbulent Dissipation Rate ......109, 114, 117 Turbulent Flow.........................................173 Turbulent Flow Past...........................54, 120

Backward Facing Step ...................54, 120 Turbulent Flows. ......................................120 Turbulent Intensities ................................118 Turbulent Kinetic Energy .........109, 114, 117 Turbulent Mixing ................. 54, 77, 120, 173

Propane........................... 54, 77, 120, 173 Turbulent Reacting Flows........................173

Linear Eddy Sub-Grid Model................173 Turbulent Reactive Flows ........................120

Free Simulations ..................................120 Turbulent Scalar Transport......................173

Linear Eddy Model ...............................173 Turbulent Viscosity ..................102, 117, 119 Turbulent Wall-Bounded Flows ...............120

Localized Dynamic Subgrid Scale Model..........................................................120

Two Fluid Modules...................................154 Two-dimensional..............................147, 276

values...................................................147 two-dimensional PDF ..............................142 Two-Fluid .........................................232, 233 two-layer-based .......................................102 U ubound.......................................................67 ucond .........................................................61 ucph_from_t...............................................61 UDEFORM_BC........................240, 246, 255 udens .........................................................25 udiff_scalar ..............................................177 UGRID .....................................................245

implement.............................................245

ui 241 underhood............................................... 185 under-relaxation..... 115, 116, 168, 182, 237,

294 Unique Filename..................................... 297 Unique Simulation................................... 297 unreacted ........................................ 133, 136 unsqueezed ............................................ 303 uouter...................................................... 314 V Vandriest Dumping ................................. 107 Vapor Mass............................................. 238 Velocity ................................................... 102 velocity sensitized................................... 102

pressure .............................................. 102 Vibration Control ..................................... 316

Damping Applications ......................... 316 Viscosity.................................... 25, 102, 116 Viscosity Evaluation Methods ................... 25 viscosity-affected .................................... 102 viscosity-affected region ......................... 102 VOF................................. 123, 232, 233, 255 Void Fraction................................... 231, 238 Volume Conditions. 25, 57, 61, 66, 108, 162,

167, 177, 209, 210, 211, 212, 216, 235, 239, 241, 243, 245, 269, 277, 278, 282, 309

Volume Flow Rate................................... 223 Volume Reaction..................................... 159 VonMises Stress ..................................... 302 Vorticity Criteria......................................... 43 W Wall Functions ........................................ 102 Wall Heat Sources ........................ 57, 67, 72 Wavelengths ........................... 206, 208, 211 Wolfstein ................................................. 102 Y YPLUS .................................. 80, 84, 85, 119 yv ............................................................ 102 Z Zeroth Order Approximation ................... 161 Zone........................................ 241, 243, 248