manual for cfd-ace software v2014

CFD-ACE+ V2014.0 Modules Manual, Part 1

©1997-2014 by ESI-Group

This ESI Group documentation is the confidential and proprietary product of ESI-Group, Inc. Any unauthorized use, reproduction, or transfer of

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CFD-ACE™, CFD-ACE+™, CFD-CADalyzer™, CFD-VIEW™, CFD-GEOM™, SimManager™, CFD-TOPO™, CFD-VisCART™, CFD-Micromesh™ and CFD-

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Revision Information

The information in this guide applies to all current ESI CFD products until superseded by a newer version of this guide.

Published July 2014

UA/CFD_/14/02/01/A

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shall not disclose the documentation and/or related know-how in whole or in part to any third party without the prior written permission of ESI

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http://www.esi-group.com/

mailto:[email protected]

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Table of Contents

Chapter 1. Modules ........................................................................................................................................ 1 Introduction............................................................................................................................................................... 1

ESI Group Around the World ................................................................................................................................ 1 About CFD-ACE+ ................................................................................................................................................... 2 About the Manuals ............................................................................................................................................... 4 Getting Started ..................................................................................................................................................... 5 Customer Support ................................................................................................................................................. 7

Flow Module ............................................................................................................................................................ 11 Introduction ........................................................................................................................................................ 11 Applications ........................................................................................................................................................ 11 Features .............................................................................................................................................................. 12 Limitations .......................................................................................................................................................... 13 Theory ................................................................................................................................................................. 14 Implementation .................................................................................................................................................. 22 Frequently Asked Questions ............................................................................................................................... 63 Examples ............................................................................................................................................................. 72 References .......................................................................................................................................................... 72

Heat Transfer Module ............................................................................................................................................. 73 Introduction ........................................................................................................................................................ 73 Applications ........................................................................................................................................................ 73 Features .............................................................................................................................................................. 75 Limitations .......................................................................................................................................................... 75 Heat Transfer Theory .......................................................................................................................................... 76 Implementation .................................................................................................................................................. 79 Frequently Asked Questions ............................................................................................................................... 99 Heat Transfer Examples .................................................................................................................................... 100 References ........................................................................................................................................................ 100

Turbulence Module ............................................................................................................................................... 101 Introduction ...................................................................................................................................................... 101 Applications ...................................................................................................................................................... 101 Features ............................................................................................................................................................ 101 Limitations ........................................................................................................................................................ 102 Theory ............................................................................................................................................................... 102 Implementation ................................................................................................................................................ 141 Post Processing ................................................................................................................................................. 154 Frequently Asked Questions ............................................................................................................................. 154 Examples ........................................................................................................................................................... 155 References ........................................................................................................................................................ 156

Chemistry .............................................................................................................................................................. 158 Introduction ...................................................................................................................................................... 158 Applications ...................................................................................................................................................... 158 Features ............................................................................................................................................................ 159 Limitations ........................................................................................................................................................ 163 Theory ............................................................................................................................................................... 163 Implementation ................................................................................................................................................ 185 Frequently Asked Questions ............................................................................................................................. 215 References ........................................................................................................................................................ 216

User Scalar Module ............................................................................................................................................... 216 Overview ........................................................................................................................................................... 216 Applications ...................................................................................................................................................... 217

CFD-ACE V2013.4 Modules Manual Part 1

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Features ............................................................................................................................................................ 217 Limitations ........................................................................................................................................................ 218 Theory ............................................................................................................................................................... 218 Implementation Overview ................................................................................................................................ 218 Frequently Asked Questions ............................................................................................................................. 225 References ........................................................................................................................................................ 226

Radiation Module .................................................................................................................................................. 226 Introduction ...................................................................................................................................................... 226 Applications ...................................................................................................................................................... 227 Theory ............................................................................................................................................................... 227 Solution Methods ............................................................................................................................................. 232 Radiative Properties.......................................................................................................................................... 263 Post Processing ................................................................................................................................................. 268 Frequently Asked Questions ............................................................................................................................. 268 References ........................................................................................................................................................ 275

Cavitation Module ................................................................................................................................................. 276 Introduction ...................................................................................................................................................... 276 Cavitation-Applications ..................................................................................................................................... 276 Features ............................................................................................................................................................ 282 Limitations ........................................................................................................................................................ 283 Cavitation-Theory ............................................................................................................................................. 284 Cavitation-Implementation .............................................................................................................................. 294 Frequently Asked Questions ............................................................................................................................. 300 References ........................................................................................................................................................ 301

Grid Deformation Module ..................................................................................................................................... 301 Introduction ...................................................................................................................................................... 301 Applications ...................................................................................................................................................... 302 Fluid-Structures Interaction Problems .............................................................................................................. 302 Simple Prescribed Motion ................................................................................................................................ 302 User Defined Motion ........................................................................................................................................ 302 Features and Limitations .................................................................................................................................. 302 Limitations ........................................................................................................................................................ 305 Implementation ................................................................................................................................................ 306 Frequently Asked Questions ............................................................................................................................. 317

Stress Module ........................................................................................................................................................ 317 Introduction ...................................................................................................................................................... 317 Applications ...................................................................................................................................................... 318 Features ............................................................................................................................................................ 324 Limitations ........................................................................................................................................................ 324 Theory ............................................................................................................................................................... 325 Frequently Asked Questions ............................................................................................................................. 330 Examples ........................................................................................................................................................... 335 Stress Concentration Demo .............................................................................................................................. 335 Hoop Stress Demo ............................................................................................................................................ 337 Large Deflection Demo ..................................................................................................................................... 338 Stress Concentration in a Circular Cylinder Validation Case ............................................................................. 339 Thermoelastic Deformation of a Cylinder Validation Case ............................................................................... 340 References ........................................................................................................................................................ 342

Electric Module ..................................................................................................................................................... 343 Introduction ...................................................................................................................................................... 343 Applications ...................................................................................................................................................... 343 Features ............................................................................................................................................................ 344 Limitations ........................................................................................................................................................ 344

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Theory ............................................................................................................................................................... 344 Implementation ................................................................................................................................................ 362 Symbol Definitions ............................................................................................................................................ 391 References ........................................................................................................................................................ 392

Appendix A Post-Processing Variables by Module ........................................................................................ 394

Appendix B Post-processing Engineering Quantities by Module .................................................................. 404 FLOW ................................................................................................................................................................ 404 HEAT ................................................................................................................................................................. 405

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Chapter 1. Modules

Introduction

ESI Group Around the World

ESI Group employs many high-level specialists worldwide. Headquartered in Paris, France, the company and its global network of agents provide sales and technical support to customers in more than 30 countries. The following figure shows some of our locations. Please visit www.esi-group.com for more locations and information.

ESI Group Locations Around the World

ESI Group Headquarters ESI Group Rungis

100-102 Avenue de Suffren

75015 Paris

FRANCE

Phone: +33 (0)1 53 65 14 14

Fax: +33 (0)1 53 65 14 12

Parc d'Affaires SILIC

99 rue des Solets

BP 80112

94513 Rungis cedex

FRANCE

Phone: +33 (0)1 41 73 58 00

Fax: +33 (0)1 46 87 72 02



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About CFD-ACE+

CFD-ACE+ is a set of software 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 - a 3D viscous, unstructured adaptive Cartesian mesh grid generation system for use with the CFD-ACE+ and CFD-FASTRAN flow solvers.

• 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.

Modules

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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.


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

• 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.)

About the Manuals

The User Manual describes the CFD-ACE+ operations and features of the CFD-ACE-Solver which are module independent. We recommend that you first read the User Manual Overview to learn the basics of how the CFD-ACE+ application works, and then review the remaining information in the User Manual.

The Modules Manual contains a section for each of the CFD-ACE+ modules that appear in the Problem Type (PT) Panel. Each section includes introduction, applications, and features sections that can help determine which module can help you model your systems.

Both the User Manual and the Modules documentation are divided into to parts. The following tables provides a quick view of where to find topics of interest.

User Manual Modules

Part 1 • Introduction

• 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

• Chimera Grid Methodology

• Appendix A - CFD-ACE+ Files

• Appendix B - DTF Utility

• Appendix C - CFD-ACE Python Scripting

• Appendix D - Physics Compatibility Matrix

Part 2


• Flow

• Heat Transfer

• Turbulence

• Chemistry

• User Scalar

• Radiation

• Cavitation

• Grid Deformation

• Stress

• Electric

• Appendix A - Post Processing Variables


• Magnetic

• Spray

• Macro Particle

Modules

5

• Introduction

• User Manual Overview

• User Subroutines

• Numerical Methods

• Mixing Plane

• Filament Model

• Heat Exchanger Model

• DSMC Method

• Applications: Electrokinetics

• Applications: Ionization

• Applications: Dielectrophoresis (DEP)

• Applications: Solidification

• Applications: Fuel Cell Modeling

• Applications: Biochemistry

• Applications: Electroplating

• Appendix A - CFD-ACE+ Files

• Appendix B - DTF Utility

• Appendix C - CFD-ACE Python Scripting

• Appendix D - Physics Compatibility Matrix

• Free Surface (VOF)

• Plasma

• Two-Fluid

• Kinetic

• Semi Device

• Appendix A - Post Processing Variables

Getting Started

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


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• 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.

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 20xx.x\UTILS\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 20xx.x\UTILS\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 20xx.x\UTILS\icons directory

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

REQUEST A LICENSE FILE

The following table gives the e-mail addresses to request a license key, or contact your local ESI Sales Representative.

Region Contact Person

Africa [email protected]

Asia Pacific

China [email protected]

Japan [email protected]

Korea [email protected]

India [email protected]

Rest of Asia [email protected]

Australia / New Zealand [email protected]









Modules

7

Europe

Eastern Europe [email protected]

[email protected]

France [email protected]

Germany [email protected]

Italy [email protected]

Spain [email protected]

United Kingdom [email protected]

Rest of Europe [email protected]

Middle East [email protected]

Americas

North [email protected]

South [email protected]

Customer Support

ESI-CFD provides excellent customer support, with staff located around the globe. Please call on us if you have any questions about the use of your software or modeling applications.

Web site http://www.esi-cfd.com (CFD Portal)

http://www.esi-group.com (Corporate)

Software support [email protected]

File attachments up to 10 MB allowed.

ftp site ftp.esi-cfd.com

Use to upload files larger than 10 MB.

Contacts USA – Huntsville +1 (256) 713-4750

USA – San Jose +1 (408) 824-1212

USA – Detroit +1 (248) 381-8040

United Kingdom +44 (0) 1543 397 900

France +33 (0)1 41 73 59 42

Germany +49 201 125 072 14













http://www.esi-cfd.com/



ftp://ftp.esi-cfd.com/


8

Italy +39 05163335577

Israel +972 77 500 5864

India – Pune +91 20 2689 8172

India – Bangalore +91 80 4017 4709

China +86 (10) 65544907/8/9/10

Japan +81 045 682 7070

+81 03 6381 8496

Korea +82 02 3660 4500/4516

+82 31 737 2987

Australia +61 2 8571 0800

Go ToMeeting http://www.gotomeeting.com / http://www.joingotomeeting.com

An e-mail invitation is usually issued, but you can look for the meeting here.

Community Forum http://www.esi-cfd.com/component/option,com_smf/Itemid,188/

Used for consulting types of questions (for example, How can I do this?, Have you ever done this?, Does anybody have an example of that?, and so forth).

Only registered members are allowed to access this section, so you will need to log in or register for an account with Community Forums.

GETTING SUPPORT FROM THE HELP MENU

From the menu bar, click Help. You can then opt to open the Help file, go to the ESI software home page, or view information about the application you are running. The About option opens a splash screen that shows version numbers for the application itself, libraries, and build platform as well as the build date and copyright information.

Note If you have not opened a model yet, you can also select the same help options from Resources on the opening window.

http://www.gotomeeting.com/

http://www.joingotomeeting.com/

http://www.esi-cfd.com/component/option%2ccom_smf/Itemid%2c188/

Modules

9

HOW TO REPORT PROBLEMS

If the Help options do not answer your questions or address your concerns, you may contact ESI Support at [email protected]. File attachments are limited to 10 MB via email. For larger files, use our ftp site (ftp.esi-cfd.com).

When you report a problem please include the following information:

• Platform information • Product/Application version number • Modules/Features you were using • Type of problem you were working on • Any error messages that you may have received in the output/log files or on the screen. • Applicable files (GGD, VGD, DTF, output files, etc. - put larger files on the ftp site) • Precise description of how you observed the problem • Instructions on how Support can reproduce the problem

TRANSFERRING FILES VIA FTP

ESI’s ftp site can handle large files, so do not be concerned about the size of DTF files that you upload.

Use your favorite ftp client or web browser to upload files to our ftp site, ftp://ftp.esi-cfd.com. The login name and password for the public account on the ftp site is listed in the Knowledge Base section of the CFD Portal.

Customers who prefer to use a web browser with drag-and-drop functionality should use the following syntax: ftp://username:[email protected]/pub

After you have accessed ftp.esi-cfd.com, you can create a directory for your files under the pub subdirectory. In your email to Support, inform Support where the uploaded files are located.

SECURITY CONCERNS WHEN USING FTP

If you are concerned about protecting proprietary data, please be aware of the following:

• You can choose to send output files, log files or scripts instead of GGD, VGD, DTF, etc. files. Also, in case of DTF files, you can choose to send only the output of the view data command:

• % DTF -vd model.DTF

This prints only the simulation data contained in the DTF file. There is no geometry or boundary condition information printed. Support can resolve many problems just by looking at the simulation data settings.

• It is ESI Group policy to handle all customer data as confidential, even if a Non-Disclosure Agreement has not been executed.

• ESI Group will sign a Non-Disclosure Agreement upon request. Contact your sales agent or Distributor to discuss this option.


ftp://ftp.esi-cfd.com/


10

• Customer Support can set up password-protected ftp accounts on the ftp site so that no other users can see your directory or files on the ftp server. No one but you and Support will know you are putting files on the ftp server. Ask Support if you wish to have a password-protected ftp account.

• Support may be able to help you using GoToMeeting. GoToMeeting enables an on-line meeting where ESI CFD Support Engineers can see your desktop, including your DTF file, and see the problems you are having without the need for you to transfer any files off of your computer. GoToMeeting can allow the remote party (Support) to take control of the mouse and drive the session, but this does not happen by default (you would have to give permission). Please request a GoToMeeting session with Support if you feel this is the best way to resolve your problem.

• You can often create a simplified model that represents your real model, but has different boundary condition values and a different geometry. Sometimes in the process of creating a simplified model to send to Support, you may discover that the simplified model works fine, and this additional information is often enough to identify the root cause of the problem.

HOW TO DETERMINE YOUR PLATFORM

If you have properly installed the ESI CFD software and configured your environment, you should be able to open a command-line shell. At the prompt, type the following text. (The subsequent output should be sufficient for Support to know what platform you are actually running.)

% python2.4_cfd >>> import platform >>> print platform.platform() >>> print platform.architecture() <Ctrl-Z> to end the session (Ctrl-D on Unix/Linux)

In addition, you should provide information about which software package you downloaded. If you are on a Unix or Linux system, or you can run Cygwin under Windows, you can get the canonical platform name by running this script:

% getCanonical.sh

If ESI_HOME is set correctly and $ESI_HOME/UTILS_2007/bin is first in your PATH, both these commands will be found. Windows users without Cygwin can send the output from the Python commands; that is sufficient.

Windows users may first need to set PYTHONHOME so that the interpreter finds its required library files. Do this either under Settings/Control Panel/System/Advanced/Environment Variables, or from a DOS shell:

% set PYTHONHOME="%ESI_HOME%\UTILS_2007\Python2.4_CFD"

HOW TO DETERMINE THE PRODUCT/APPLICATION VERSION NUMBER

1. Launch the product/application (CFD-GEOM, CFD-ACE-GUI, etc.)

2. Click the Help menu.

3. Click About <application name> (for example, About CFD-ACE-GUI).

4. Make a note of the version number.

-or-

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1. Open a command prompt.

2. On the command line, enter, for example, CFD-ACE-GUI -v and press Enter. This command is case sensitive and includes a space before –v. A file (CFD-ACE-GUI.version in this case) is created, in the current working directory, which contains the build date and version information.

Flow Module

Introduction

This sectiontopic describes the Flow module which is the heart of CFD-ACE+ and is used in most simulations.

Use the Flow module to find the solution for (1) the velocity field by solving for the x-, y-, and z-momentum equations and (2) the pressure field by solving the pressure correction equation.

You can use the Flow module with one or more of the other CFD-ACE+ modules to provide a multiphysics-based solution to an engineering problem. (For example, you can couple flow with heat transfer, mixing, finite-element stress solution, and so forth.)

You can read more about the Flow module in the following topics:

Applications Features Limitations Theory Implementation Frequently Asked Questions Examples References

Applications

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. Following 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 the Flow module to produce a multiphysics simulation. A laminar flow is assumed unless the Turbulence module is activated.

Flow Visualization

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


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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. In addition, CFD-ACE+ can determine the mass flow rate through a system for a given differential pressure. Also, mass flow calculations are useful for determining flow splits when the flow is bifurcated.

Multiphysics Applications

You can use the Flow module with many of the other CFD-ACE+ modules to perform multiphysics analyses. The more commonly included modules are listed here. Examples of these types of applications are given in each module’s Examples 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) (see Module Manual, Part 2) Free Surfaces (Flow is required) (see Module Manual, Part 2) Two Fluid (Flow is required) (see Module Manual, Part 2) Cavitation (Flow is required) Grid Deformation Finite Element Stress Plasma (Flow is required) (see Module Manual, Part 2) Kinetic (see Module Manual, Part 2) Electric and Magnetic Module (Electrophysics)

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.)

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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 (that is, 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 model (MO > Adv > Slip Walls) is included that allows for velocity slip and a temperature jump at the walls. (See Theory > Slip Walls for details about this model.)

If Slip Walls is activated, then the Slip Model boundary condition is enforced at all walls and the solid-fluid Interfaces by default. You can, however, also set the No-Slip boundary condition for each wall separately. This capability allows you to apply the Slip Model and the No-Slip condition on a wall-by-wall basis. The Slip Walls model can also be applied for Rotating Walls and Solid-Fluid Interfaces.

Hemolysis Model

Shear stress exerted on blood may damage or destroy red blood cells. The phenomena of destruction of red blood cells and subsequent release of hemoglobin is called hemolysis. This phenomena commonly occurs 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 in 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. In addition, the subthreshold damage to the cells may accumulate over the time and account for delayed hemolysis. To compute a mass-averaged hemolysis index, an explicit Lagrangian type particle tracking scheme is developed in CFD-ACE+ as a post processing tool. 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 such as CFD-FASTRAN.


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Theory

FLOW MODEL THEORY

This sectiontopic describes the mathematical equations used by the Flow module. For details on the methods used to solve these equations, see CFD-ACE+ User Manual, Part 2 > Numerical Methods.

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

• The mass of a fluid is conserved; that is, 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).

CFD-ACE+ uses these two laws to develop a set of equations, known as the Navier-Stokes equations, to solve numerically using an iterative method See the following for more information.

Mass Conservation Momentum Conservation Navier-Stokes Equations Simple Flow Model Theory Slip Wall Theory

MASS CONSERVATION

Conservation of mass (Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002.) 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 equation can be expressed as:

(1)

The first term on the left 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

Newton’s second law (Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002.) 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 force

• Viscous force

Body forces

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• Gravity force

• Centrifugal force

• Electromagnetic force

• Surface tension force

• Momentum resistance

• Porous media forces

Surface Forces

Here we describe surface forces. Body forces are included as source terms, and are discussed in CFD-ACE+ User Manual, Part 1 > Rotating Systems (gravitational and rotational body forces ). Also, the Magnetic module provides 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:

A

(2)

Similar equations can be written for the y- and z-components of the momentum equation. (In these equations, ρ is the static pressure and τij is the viscous stress tensor.)

(3)

(4)

NAVIER-STOKES EQUATIONS

The momentum equations ((2, 3, and 4), contain as unknowns the viscous stress components τij; therefore, a model must be provided to define the viscous stresses (Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002).

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:

(5)


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(6)

(7

(8)

(9)

(10)

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

(11)

(12)

(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:

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(14)

(15)

(16)

For more details on the discretization of these equations and the method used to obtain velocity-pressure coupling, please see CFD-ACE+ User Manual, Part 2 > Numerical Methods.

SIMPLE FLOW MODEL THEORY

The Simple Flow model in CFD-ACE+ assumes that the velocity profile perpendicular to a wall boundary is given by the parabolic velocity profile of laminar flow theory for fully developed flow. This profile is internally assumed or imposed within each cell that has a face lying in a wall boundary that is identified or designated by the user to be a simple-flow boundary.

The complete velocity profile under the laminar, fully developed flow assumption is fully parameterized by the velocity at the centroid of the boundary cell. Thus, every needed property related to the velocity profile, including the shear rate and the shear stress at the wall, is fully determined from the velocity at the cell centroid. The shear stress at the wall is then used in the momentum equation in the usual way, just as it would have been used if the shear stress at the wall were computed by any other means.

Effectively then, the Simple Flow model assumes a local velocity profile from which the needed shear stress can be deduced without having to use multiple computational cells to resolve the velocity profile, and the shear stress, at the wall. This means that the pressure drop or flow rate along a wall boundary can be determined using this model with only a fraction of the number of computational cells needed for a fully resolved computation. On the other hand, the greater the deviation of the real velocity profile from the assumed parabolic velocity profile of laminar flow theory, the greater the loss of accuracy compared to a fully resolved computation, which in the limit is assured to be accurate.

As an example of the application and compromises made in using the Simple Flow model, consider the case of axial flow in a duct with a rectangular cross-section. If the flow is laminar, it will be necessary to have at least, say, eight cells along each edge of the cross-section of the duct to resolve the velocity profile and accurately predict the corresponding pressure gradient or flow-rate along the duct. On the other hand, with the Simple Flow model, a single cell can represent each axial section of the duct, leading to a reduction in the number of cells (relative to the resolved, 8 x 8, computation) by a factor of 64. However, if the flow is not laminar, or if the edge or corner effects in the flow are appreciable, then the use of the simple flow model could result in significant deviations from the actual physical problem, and from the corresponding full-resolution computation.

As can be inferred from this discussion, a typical attractive use of the Simple Flow model would be for modeling flow networks, including micro-channel system, where the Reynolds Number is sufficiently low to ensure that the


18

flow will be strictly laminar, and where geometric complexity or the additional physical models that are active in the simulation require the number of cells to be kept to a minimum, the more so if the simulation is transient with a small time scale.

The Simple Flow model in CFD-ACE+ is implemented in two different variants: one-cell wall model and second-order wall model.

SIMPLE FLOW MODEL FEATURES

The Simple Flow model is primarily intended for fully developed laminar flow between a pair (or two pairs) of walls that are facing each other. The facing walls need not be parallel to each other. The walls can be straight or curved and can be moving as well. Examples of flows which can be simulated are fully developed laminar flows in channels of rectangular cross-section. This model can also be used to calculate the wall shear stresses in laminar boundary layers when it is not affordable to resolve the boundary layer. For fully developed laminar flow between two parallel plates, the predictions of the Simple Flow model are exact.

SIMPLE FLOW MODEL LIMITATIONS

• The One Cell Wall option is supported for grids which can be created by perpendicular extrusion. For example, extruding lines to create quadrilaterals in 2D or extruding triangles or quadrilaterals to create prisms or hexahedra, respectively, in 3D. For grids which cannot be created by perpendicular extrusion, the model may still work but there will be some loss of accuracy.

• For the One Cell Wall option, the velocity in CFD-VIEW will appear to be wall velocities (or zero). This is due to the fact that CFD-VIEW outputs values at the nodes of the cells and all the nodes lie on a wall boundary (as shown in figure 1(a)), and for stationary walls the velocity for no-slip conditions is zero . This is a limitation of visualization that the velocities calculated by CFD-ACE+ cannot be displayed. The results can be verified by visualization of pressure distribution, density, and so forth, which can all be viewed in the usual way.

• This model assumes fully developed flow which is true beyond the entrance length for a given flow. An approximate correlation for entrance length is given by (Shaw and London. 1978) as Entrance Length = Dh (0.5 + 0.5 Re) where Re is the Reynolds number based on mean flow velocity, Dh = 4 A / lp is the hydraulic diameter, A is the cross-sectional area, and lp is the wetted-perimeter.

• This model assumes a parabolic velocity profile locally perpendicular to the wall boundary. This parabolic velocity profile coincides exactly with the analytical solution for fully-developed laminar flow between two infinite parallel plates. Therefore, the predictions of the Simple Flow model are exact for such a case, and are highly accurate for channels of rectangular cross-section with high aspect ratio. For channels of rectangular cross-section with low aspect ratio, using a parabolic velocity profile for rectangular ducts under-predicts the pressure gradient for a given mass-flow rate or the mass-flow rate for a given pressure gradient. The percentage error in pressure gradient as a function of the aspect ratio as shown in following figure. The error increases as the aspect ratio of the duct increases and is maximum for square cross-sections.

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Percentage Relative Error (ΔPanalytical - ΔPSFM / ΔPanalytical) as a function of aspect ratio in using parabolic velocity profile for channels of rectangular cross-section

• There is a loss of accuracy if the flow is turbulent or transitional.

• There is a loss of accuracy for unsteady flows. The stronger the time dependence of the flow, the greater the loss of accuracy.

• If the walls are not parallel to each other but are inclined at an angle + a, as shown in the following figure, the velocity profile at a given cross-section is parabolic if |α Re|<<1, where Re is based on the local maximum velocity and the local distance between the two plates at the cross-section. The shape of the velocity profile for three different values of α Re is shown in the following figure. For converging channels (α Re < 0), the velocity profile deviates from the parabolic shape and becomes flatter as α Re decreases. For diverging channels (α Re > Re 0), when α Re > ≈ 10 a back-flow region may develop, as shown in the following figure.

• When using the Second Order Wall option, the center of the cell adjacent to the wall should lie outside the boundary layer because this model assumes this cell-center velocity to be the free-stream velocity. This model does not account for the variation of boundary layer thickness with distance (such as over a flat plate) and the grid should be constructed such that the size of the cells adjacent to the wall is greater than the local boundary layer thickness.


20

Dependence of velocity profile between two infinite plates as a function of α Re

VISUALIZATION OF RESULTS

It is not possible to visualize the assumed parabolic velocity profile in CFD-VIEW. CFD-ACE+ only outputs values at the nodes of cells, it does not output interpolated values that show variation within individual cells.

For the One Cell Wall option, the velocity in CFD-VIEW will appear to be the wall velocities (or zero). This is due to the fact that all the nodes lie on a wall boundary, and for stationary walls the velocity is zero for no-slip conditions. This is a limitation of visualization that the velocities calculated by CFD-ACE+ cannot be displayed. The results can be verified by visualization of pressure distribution, density, and so forth, which can all be viewed in the usual way.

IMPLEMENTING SIMPLE FLOW MODEL

The Simple Flow model includes three options:

1. High Order Wall Local - allows you to choose a subset of walls in the geometry for options 2 and 3.

Note Options 2 and 3 apply globally to all walls in the geometry.

2. Second Order Wall Global - applies the second order wall model globally; that is, it applies to all walls in the geometry. This option models the wall shear stresses in laminar boundary layers when you do not want to resolve the boundary layer. Multiple cells can be next to the wall for which this option is specified, as shown in figure (c).

3. One Cell Wall Global - applies one cell wall model globally; that is, to all pairs of walls facing each other in the geometry. This option models the shear stress at walls by assuming a parabolic velocity profile between the two pair of walls on which this option is specified. By applying this option to a rectangular cross-section, two parabolic velocity profiles—one for each pair of walls—will be imposed. There can be only one cell between the two pairs of walls on which this option is specified as shown in figure (a).

Suitable grid configurations for different Simple Flow Model options where does figure b come in?

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ONE-CELL WALL OPTION

This option is intended to model the effects of a parabolic velocity profile between two walls facing each other. The shear-stress at the walls is calculated assuming a parabolic profile for velocity

USFM = ay2 + by + c (17)

where a, b, and c are constants and y is the local distance normal to the wall. The parameters a, b, and c are such that the velocity profile satisfies the following three boundary conditions:

USFM, 1 = Uwt1

at wall 1, where Uwt1 is the tangential velocity of wall 1, and

USFM, 2 = Uwt2

at wall 2, where Uwt2 is the tangential velocity of wall 2.

CFD-ACE+ solves the Navier-Stokes equations to obtain the cell volume average value of velocity. The assumed velocity profile satisfies the condition where is the cell volume average value of velocity and is obtained from the standard numerical solution process, USFM is the assumed velocity profile, and V is the cell volume.

By applying this option to a rectangular cross-section, two parabolic velocity profiles (one for each pair of walls) is imposed.

SECOND-ORDER WALL OPTION

The Second Order Wall option represents the effect of boundary layers attached to wall boundaries. This option differs from the One Cell Wall option in that the velocity profile in (17 is used.

Instead of applying (17 between two opposite walls, the Second Order Wall option applies the equation only in the cell adjacent to the wall. The relevant boundary conditions satisfied by the velocity profile are as follows:

USFM,1 = Uwt at the wall

USFM,2 = U at the cell center

∇ USFM = ∇ u at the cell center

where

Uwt is the tangential wall velocity, and

U is the velocity obtained from the standard numerical solution process.

SLIP WALL THEORY

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


22

(18)

where

α = Accommodation coefficient (user input)

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

σ = Molecular diameter (hard wired value = 3.579e-10 m)

NA = Avagadro's number = 6.023e26 atoms/kmol

R = Gas constant = 8314 J/kmol-K

Tic = Local temperature (K)

Pabs = Pic + Pref = Local absolute pressure (Pa)

= Normal velocity gradient at the wall

Temperature slip, (assuming Ts and Tw are the slip and wall temperatures, respectively) is

(19)

where

= Normal temperature gradient at the wall

For more information about this model, see Features and Limitations of the Flow module.

Implementation

IMPLEMENTATION AND GRID GENERATION

The Implementation and Grid Generation section provides details about setting up a model for simulation using the Flow module. This section includes the following topics:

Problem Type Model Options Volume Conditions Boundary Conditions Initial Conditions Solver Control Settings

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Output Options Post Processing

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

The general grid generation concerns apply, that is, ensuring that the grid density is sufficient to resolve solution gradients, minimizing skewness in the grid system, and locating 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 (such as the bend of a pipe).

PROBLEM TYPE

The Flow module is required for most simulations and can be coupled with virtually all other modules. To activate the Flow module, click the Problem Type [PT] tab to open the Problem Type panel. Then check the box next to Flow.

Note The CFD-ACE+ User Manual contains more information on using the Control Panel and other panels.

MODEL OPTIONS

To access model option panels, click the Model Options [MO] tab. With only Flow selected in the problem type, three side tabs are available on the MO panel:

Shared Flow Advanced

SHARED TAB

Under the Shared tab, all of the following options can be selected or specified:

Panel Option Description

Simulation Description Title

The title is optional and can be chosen entirely at the discretion of the user.

Polar (Axisymmetry about X-axis)

Choose Non Axisymmetric or Axisymmetric. The axisymmetry determines whether a two-dimensional geometry is treated in an axisymmetric formulation or in planar formulation in the solver. This option only appears for problems with two-dimensional geometries. For all axisymmetric cases, the model must be set up by the user such that the x-axis corresponds to the axis of symmetry and such that the grid lies entirely in the first quadrant.

Simulation time- This time must be chosen from among one of two options: (i) steady or (ii) transient.


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dependence For the transient option, additional inputs and specifications identifying the time-stepping control and the time-integration scheme must be provided.

Body Forces If the gravity option is activated, the gravitational force vector must be specified.

If the effects of gravity are to be modeled using complete simulation, then a reference density must also be specified. If, on the other hand, only the thermally induced buoyancy effects of gravity are to be approximated using the Boussinesq Approximation, then the desired reference temperature and the volumetric coefficient of thermal expansion of the fluid must also be specified.

If the simulation contains two immiscible fluids, then two reference temperatures and two volumetric coefficients of thermal expansion, one pair of values for each fluid, must be specified.

Additional information on choosing the reference density is provided in the FAQ section of the Flow Module.

Rotational reference If a rotational reference frame is chosen, two options are available: (i) volume-condition based rotating frames and (ii) a global (absolute) rotating frame. The volume-condition based option allows multiple reference frames to be used, up to one frame for each volume condition in the model. The global (absolute) option performs all calculations in the single rotating frame of reference. Additional information on setting the rotation reference is provided in the CFD-ACE+ User Manual > Rotating Systems chapter.

Chimera If the chimera grid module is activated, then the additional chimera settings must be provided in the VC and BC tabs. Additional information on the Chimera Grid option is provided in the CFD-ACE+ User Manual > Chimera Grid Methodology chapter.

FLOW TAB

The flow tab lets you set specific reference pressure and various simulations.

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Model Options in Flow Module Settings Mode


Pressure 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 (for example, 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 gauge 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.

Fan Model For fans not defined in a BC.

Virtual Resistance Model For momentum resistance, porous media, or heat exchangers not defines in a VC


26

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

(20)

where

τ = 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.

ADVANCED TAB

Use the Advanced tab to specify specific types of simulations. The main options include Flow and Primary Fluid.


Simple Flow Models Check the Simple Flow Model box to activate one of these models. There are three

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options available:

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

3. 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.

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]

(21)

where

μ = gas viscosity a and

b = constants

Kn = Knudsen number

Primary Fluid This option allows you to select a Fluid Properties file (Properties File Name > Browse) containing tabulated data (ASCII), from which some fluid properties (such as viscosity) can be evaluated.

Activation of this evaluation method for a particular fluid property is performed under the VC panel, within the corresponding property tab, and on a VC per VC basis. For example, if one wants to use the Fluid Properties file to evaluate the viscosity in some VC, simply go to the VC > Fluid tab and select Read From Fluid Properties File for viscosity.


28

Activation of the “Fluid Properties File” option for viscosity for the selected VC

Fluid Type Three Fluid Type options are available, each requiring a different data format. The options include Real Fluid, Simplified Liquid, and Perfect Gas.

Fluid Options under the Flow Section

Real Fluid

With this option, the following properties are tabulated as a function of temperature and pressure: density, specific volume, specific enthalpy, specific entropy, specific energy, specific heat capacity at constant pressure, specific heat capacity at constant volume, isentropic exponent, thermal volumetric expansion coefficient, isothermal compressibility, speed of sound, thermal conductivity, and viscosity.

The value of the property at a given temperature and pressure is calculated using the

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bilinear interpolation.

Simplified Liquid With this option, the following properties are tabulated as a function of temperature: heat capacity at constant pressure, viscosity, thermal conductivity, density, speed of sound, and vapor pressure. There is no dependence of these properties on pressure.

The value of the property at a given temperature is calculated using basic linear interpolation

Perfect Gas

With this option, the following properties are tabulated as a function of temperature: heat capacity at constant pressure, viscosity, and thermal conductivity. There is no dependence of these properties on pressure.

The value of the property at a given temperature is calculated using basic linear interpolation.

Note As of V2013.0, not all tabulated properties are used: only the following can be evaluated via the Fluid Properties file for the primary fluid: density, viscosity, specific heat capacity at constant pressure, and thermal conductivity.

DATA FORMATS FOR FLUID PROPERTIES

The data formats for the Fluid Properties input file are as follows:

Real Fluid

For a Real Fluid, the properties must be arranged in columns, but the format is slightly different from the Perfect Gas and Simplified Liquid because it must account for temperature and pressure dependence.

The general format for the Real Fluids input file is a sequence of temperatures tables, each containing rows of data as a function of temperature, and where each temperatures table corresponds to a given pressure:

p1

T1 dens1 Cp1 ...

T2 dens2 Cp2 ...

... ... ... ...

Tn densn Cp3 ...


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p2

T1 dens1 Cp1 ...

T2 dens2 Cp2 ...

... ... ... ...

Tn densn Cp3 ...

The properties tabulated in the real fluid input file are listed in the following table:

Real Fluid Properties Table Property Unit

Pressure Pa

Temperature K

Density kg/m3

Specific volume m3/kg

Specific enthalpy J/kg

Specific entropy J/(kg.K)

Specific energy J/kg

Specific heat capacity at constant pressure J/(kg.K)

Specific heat capacity at constant volume J/(kg.K)

Isentropic exponent -

Thermal volumetric expansion coefficient 1/K

Isothermal compressibility 1/Pa

Speed of sound m/s

Thermal conductivity W/(m.K)

Viscosity Pa.s

The detailed file format for Real Fluid, taken from EcosimPro’s ESPSS Library user manual, is described here:

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Line no. Description

1 Text line for gas properties (for example: Gaseous Properties)

2

One line with the following values (free number format), in that order:

Molecular weight (g/mole), Ref. pressure (Pa), Ref. temperature (K), Ref. entropy

(J/kg.K), Ref. enthalpy (J/kg.), Minimum value of pressure (Pa), Maximum value of

pressure (Pa), Minimum value of temperature (K), Maximum value of temperature (K).

3 to 5

A line (free number format) with the properties of Real Fluid Properties Table above for

the triple point (liquid).

Another line for the triple point (vapor).

Another line for the critical point.

6 One line with the number of pressures (npG) in gas conditions.

7 One line with the first pressure value, followed by the number of temperatures (nTG1)

considered for this particular pressure.

8 to 8 + nTG1

For every considered temperature, one line with all the properties (see note 1).

The temperature range must go from the saturation point (or the melting point in

supercritical conditions) to the maximum temperature.

9 + nTG1

to

9 + npG + ΣnTGi

For every pressure, include a new temperatures table like before (lines 7 and following).

The pressures must go from one below or equal to the triple point to another higher

than the critical point. The triple and the critical pressures must be included.

For pressures lower than the triple one, temperatures must go from the minimum

(melting) to the maximum.

For pressures going from the triple one to the critical one, temperatures must go from

the saturation temperature to the maximum. For pressures equal or higher than the

critical one, temperatures must go from the minimum (melting) to the maximum (see

note 2)

1 Text line for liquid properties (for example: Liquid Properties). This line must follow the

last gas line.

2 One line with the number of pressures (npL) in liquid conditions


32

3

One line with the first pressure value (the triple point), followed by the number of

temperatures (nTL1) considered for this pressure, and finishing with the surface tension σ

(N/m) at the saturation temperature of the current pressure.

4 to 4 + nTL1 For every considered temperature, one line with all the properties (see note1). The

temperature range must go from the melting point to the saturation temperature.

5 + nTL1

to

9 + npL + ΣnTLi

For every pressure (including the critical one), include a new temperatures table like

before (line 3 and following).

The technique used to generate a liquid table consists of the following: For the triple

pressure, only one temperature point is considered; two temperatures (melting and

boiling point) for the second pressure; add one more point (that of the new saturation

temperature) for the next pressure, etc.

The pressures must go from the triple point to the critical point, and must have the same

values as in the gas tables. The triple and the critical pressures must be included.

Note The properties stored (columns) are those given in Real Fluid Properties Table beginning with temperature and in the same order. The pressure is stored only at the beginning of every temperatures table. The temperatures considered for a given pressure can be different from the ones considered for another pressure. The interpolation procedures are not restricted to rectangular tables.

Simplified Liquid

For a Simplified Liquid, the properties must be arranged in columns, with one row per temperature. For example:

Liq_Prop_N2O4 92.01

Temp dens Cp vsound cond visc Pvap

258.1 1523 1438.6 1163.6 0.144 0.00061 15200

262 1514 1458.2 1147.1 0.143 0.00059 19000

...

...

333.1 1353 1733 831.8 0.106 0.00026 506000

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Note Words in bold are keywords. They must be present in the file exactly as written here and in that order (and, therefore, the properties must follow that order as well. The numeric value of the first line is the molecular weight. The format for numbers is free.

Units Property Unit

Temp K

dens kg/m3

Cp J/(kg.K)

vsound m/s

cond W/(m.K)

visc Pa.s

Pvap Pa

Perfect Gas

For a Perfect Gas, the properties must be arranged in columns, with one row per temperature. For example:

Gas_Prop_Air 28.95

Temp Cp visc cond

200 1004.37 1.437e-005 0.01992

208 1004.37 1.4653e-005 0.02032

...

...

1000 1141.5 4.1773e-005 0.06720

Note Words in bold are keywords. They must be present in the file exactly as written here and in that order (and therefore the properties must follow that order as well). The numeric value of the first line is the molecular weight. The format for numbers is free.

Units


34

Property Unit

Temp K

Cp J/(kg.K)

visc Pa.s

cond W/(m.K)

VOLUME CONDITIONS

Click the Volume Conditions [VC] tab to see the volume conditions panel. 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.

Tip See User Manual > Control Panel > Volume Conditions for details on this panel.

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 in the ACE+ User Manual.

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.

Required Volume Condition Properties 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

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) Mix Piecewise Linear in T Read from Fluid Properties File

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Read from Fluid Properties File

VOLUME CONDITION DENSITY PROPERTIES

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, use the Ideal Gas Law. The Ideal Gas Law is given by

where

p ref = reference pressure r = calculated static pressure

MW = species or mixture molecular weight R = universal gas constant T = temperature.

POLYNOMIAL IN T

The Polynomial in T option calculates the density as a function of temperature using a polynomial.

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

PIECEWISE LINEAR IN T

The Piecewise Linear In T 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 Required Input(s) Data pairs of Temperature and Density


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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, Chemisty 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.

MIX PIECEWISE LINEAR IN T

The Mix Piecewise Linear in T option calculates 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, Chemisty 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 ρv is the vapor density and rl is the liquid density. 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.

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USER SUBROUTINE (UDENS)

The User Subroutine (UDENS) option implements 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 fluid. 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.

READ FROM FLUID PROPERTIES FILE

With this option, the fluid density will be interpolated from the Fluid Properties data file specified under the MO/Adv tab under the Primary Fluid section. For more information, please refer to the section Flow Module > Implementation > Model Options > Advanced Tab > Primary Fluid.

VOLUME CONDITION VISCOSITY PROPERTIES

CONSTANT (KINEMATIC)

The kinematic viscosity is given as follows

where μ is the dynamic viscosity and ρ 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 ρ is the density of the fluid and ν 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 of B is 112K. These values are for air at moderate temperature and pressures.

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


38

POLYNOMIAL IN T

The Polynomial in T option is given as follows

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


The Piecewise Linear in T option linearly interpolates 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

Mix Kinetic Theory uses the kinetic theory of gases to calculate the viscosity of the gas or mixture of gases (Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002). The viscosity is defined as follows

where

μ

i

= dynamic viscosity of species i

MWi = molecular weight of species i T = temperature in Kelvin σi = characteristic diameter of the molecule in Angstroms

Ωμ = collision integral.

The collision integral, Ωμ, is given by

where T* is the dimensionless temperature and is given by

and where

ε = characteristic energy k = Boltzmann's constant

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T = temperature

To calculate the mixture viscosity using kinetic theory, the following equation is used

where

xi ,xj = molar fraction of species i and species j μi = viscosity of species i Φij = dimensionless quantity

and Φij 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 (Liquid)


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

MIX POLYNOMIAL IN T (LIQ)

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 μi = viscosity of species i calculated using a Polynomial in T.

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

POWER LAW

This option uses a non-Newtonian Power Law [Bird, et al., 2002] model to calculate the viscosity of the fluid. The Power Law model is

where

and

μo = the zero shear rate viscosity K, A1, A2, A3, A4, B = constants characterizing the fluid

N = the power law index T = temperature

D0 = the cutoff shear rate D = the local calculated shear rate.

For a temperature-dependent viscosity, A1 or A2 needs to be a non-zero value. If A1, A2, A3, A4, µ0, and D0 are set to zero, then the simplest form of the Power Law model is recovered, which is the two-parameter power law (also know as the Ostwald-de Waele Model [7]) given by:

The Power Law index determines the classification in which the fluid falls:

N = 1 indicates the fluid is Newtonian N >1 indicates a shear thickening fluid (dilatant fluid) N < 1 indicates a shear thinning fluid (pseudo-plastic)

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Required Module(s) Flow Required Input(s) μ0, N, D0, K, A1, A2, A3, A4

CARREAU LAW

This option uses the Carreau Law (Carreau, P. J. 1968) model to calculate the viscosity of the fluid. The Carreau Law model is

where

μo = the zero shear rate viscosity μ∞ = the infinite shear rate viscosity

N = the power law index T = temperature a = constant

= the local calculated shear rate K = 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) μ0, , μ∞, n, K, a

POWER LAW (BLOOD)

This model (Ballyk, P.D., D.A. Steinman, and C.R. Ethier. 1994) is available when solving for Flow and the subtype of the fluid is liquid. The model is:

and

γ = the local calculated shear rate

l = the consistency constant

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

Δμ = 0.25 (Default)

a = 50 (Default)


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b = 3 (Default)

c = 50 (Default)

d = 4 (Default)

Δn = 0.45 (Default)

n∞ = 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) μ∞Δμ, n∞, Δn, a, b, c, d

CASSON MODEL

This model (Fung. 1993) is available when solving for Flow and the fluid subtype is liquid. The model is

where

and γ = the local calculated shear rate

τy = the yield stress in shear given by τy= (0.0625Hct)3 Hct = the blood hematocrit and should be specified as a fraction between 0 and 1

η = a constant η0 = the viscosity of the plasma.

We define and .

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) μ∞, η∞

WALBURN AND SCHNECK

This model (Walburn and Schneck. 1976) is available when solving for Flow and the fluid subtype is liquid. The model is

where

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γ = the local calculated shear rate

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

Hct = the blood hematocrit and should be specified as a 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

READ FROM FLUID PROPERTIES FILE

With this option, the fluid dynamic viscosity will be interpolated from the Fluid Properties data file specified under the MO/Adv tab under the Primary Fluid section. For more information, please refer to the section Flow Module > Implementation > Model Options > Advanced Tab > Primary Fluid

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 the following topics:

Inlets Outlets Walls Rotating Walls Symmetry Interfaces Thin Walls


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Cyclic Periodic

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 several methods (subtypes) available in CFD-ACE+:. In addition, all of the inlet boundary condition subtypes allow for velocity directions to be specified in various ways. Fixed Velocity Fixed Mass Flow Rate Fixed Total Pressure Fixed Pressure Fan

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:

(22)

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:

(23)

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FIXED TOTAL PRESSURE

Fixed Total Pressure, an inlet subtype, allows you to fix the total pressure (P0) and total temperature (T0) at the boundary patch (Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002). For ideal gases, the total temperature and pressure are computed using:

(24)

(25)

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

(26)

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.

FAN

Inlet fan is a total pressure inlet with pressure jump and torque applied to the cells next to the inlet. The thrust and torque can be specified in several different ways.

1. A constant value for pressure jump and torque can be specified.

Δp = constant

τ = constant


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2. The pressure jump and torque can be specified as a 5th order polynomial of the local normal velocity through the fan surface.

The user can limit the above polynomial functions to a range of normal velocities and . There is also an option to specify pressure jump and torque as a function of the average normal velocity going through the entire fan surface.

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3. A profile of volumetric flow rate through fan surface versus pressure jump and torque can also be specified in the way it is done for the FAN model.


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4. A file containing the volumetric flow rate through fan surface versus pressure jump and torque can also be specified in the way it is done for the FAN model.

For the input file option, the data must be given in the following format:

Line 1: Number of Points (N) in the Profile Input <Integer Value>

Lines 2 - (N + 1): Volume Flow Rate - Pressure Head - Torque Shear Stress <3 Real Values>

Note that the third column (Torque Shear Stress) needs to be provided only when the swirl option is activated. An example file would be:

3

10 10 10

20 20 20

30 30 30

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The user is present with a piecewise linear or a spline fit thorough the above specified profile or file data. To specify fan direction, the user has to pick a volume condition towards which the fan should provide the thrust.

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, and Swirler.

Velocity direction mode Description 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.

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

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.

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.

Inlet Boundary Condition Subtypes and Variables Subtype Required variables 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


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Fixed Total Pressure (Normal) Po, To, Direction, Thrust, Torque

Fixed total Pressure (Cartesian) Po, To, Direction, Nx, Ny, [Nz]3D,2Ds, Thrust, Torque

Fixed Total Pressure (Cylindrical) Po, To, Direction, Va, Vr, [Vt]3D,2Ds, Thrust, Torque 3D Available for 3D simulations. 2Ds Available for axisymmetric 2D swirl simulations.

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.

Notes Inflow through an outlet can occur anytime during the solution convergence process, even if the final solution indicates all outflow. Therefore, we suggest 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 a nonphysical 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.

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.

The following outlet methods (subtypes) are available:

Fixed Pressure Farfield Fixed Velocity Extrapolated Fan

Outlet Subtype Description

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

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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.

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.

Fan Outlet fan is a fixed pressure boundary with pressure jump and torque applied to the cells next to the outlet. More details about thrust and torque input can be found in inlet BC section.

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

Fan P, Direction, Thrust, Torque T

3D Available for 3D simulations.


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2Ds Available for axisymmetric 2D swirl simulations.

WALLS

Wall boundary conditions allow the specification of wall velocities. The required flow variables are U, V, W or Omega (W being present only for 3D and 2D swirl simulations and Omega present only in 2D swirl simulations). The Wall boundary condition for the flow module includes the following subtypes:

Walls Subtype Description

No-Slip The No-Slip Wall BC subtype is the default setting fro all flow cases except when Slip Walls Model is activated (under the MO tab). The imposed boundary conditions are

where is the velocity vector of the fluid in contact with the wall, and is the velocity vector of the wall. For stationary walls, U = V = W = 0. For moving walls, the velocity values should be specified.

Inviscid

The Inviscid wall BC subtype is available for all flow cases except when Slip Walls Model is activated (under the MO tab). The imposed boundary conditions are

where is the normal component of the velocity vector of the fluid in contact with

the wall, and is the normal velocity vector of the wall. This BC places no limitation

on the tangential components of the velocity vector of the fluid. This is very similar to a symmetry BC except that normal wall velocities (with grid deformation) and boundary conditions for other modules (such as heat, electric, and so forth.) can now be applied on this wall. The inviscid behavior is applied only for the flow module.

Slip Model The Slip Model subtype is only available when the Slip Walls Model is activated (under the MO tab). If the Slip Walls Model is activated, this BC subtype is applied to all Walls and Solid-Fluid Interfaces by default. However, the user may change this default assignment as explained in Theory-Slip Walls. For the most part, Solid Fluid Interfaces are also treated as Walls. If the High Order Wall Local simple flow model has been activated (see Model Options-Advanced Tab) then the user will have the opportunity to select which reduced flow model to apply to the selected wall. The choices are No Wall Model, One-Cell Wall, and Second-Order Wall.

By default, the first-order slip (Maxwell) model activates when you specify the Slip Wall model. You also have the choice of defining User defined slip on the same tab. In this case, you will be able to define your own slip model which will be applicable to the walls and solid-fluid interfaces.

ROTATING WALLS

A rotating wall boundary condition can be used to set a rotational velocity profile on a wall. Rotating Walls have the same three subtypes (No-Slip, Inviscid, and Slip Model) as regular Walls. The required variables for the Flow Module are Cx, Cy, Cz (the x, y, z location of any point on the axis of rotation), and Wx, Wy, Wz (a vector that defines the rotation direction and the magnitude).

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If the High Order Wall Local simple flow model has been activated (see Model Options-Advanced Tab) then the user will have the opportunity to select which reduced flow model to apply to the selected wall. The choices are No Wall Model, One-Cell Wall, and Second-Order Wall.

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.

INTERFACES

From 2013.0 version onwards, interface has also subtypes and now user can apply a fan (pressure difference) on a fluid-fluid-interface. Interface has following sub-types: Interface and Fan.

Interfaces Subtype Description

Interface 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.

Fan The fan can be specified at fluid-fluid interface by supplying thrust and torque. Please refer inlet BC section for more details of thrust and torque input. Users must ensure that to specify fan BC at the interface the fan location must be boundary of the two separate volumes (fluid –fluid interface).

Interface Boundary Condition Subtypes and Variables Available Subtypes Required Variables Optional Variables

Interface

Fan Direction, Thrust, Torque

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.

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.


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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.

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.)

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.

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.

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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.

ADVANCED SETTINGS

Advanced Settings Description

Shared Buffered Output

Higher Accuracy

Minimum Face Angle for Skew Term

Ignore Angle Below __ deg

Flow The Advanced options tab under the Flow heading includes Cut Diffusion (Flow) and CFL Relaxation.

The Cut 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.


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Note Rule of Thumb, the inverse value of the usual inertial relaxation factor.

Effect of Value

5 = Default Value

1 = More stability, Slower convergence

100 = Less stability, Faster convergence

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

OUTPUT OPTIONS

The output desired from the Flow Module can be specified under the Out (Output) tab in the ACE+ GUI. In terms of type and usage, the data that can be output from the Flow Module falls into five different categories: (i) restart data; (ii) graphical solution data; (iii) summary data; (iv) monitor point/plane data; and (v) user-specified data. Restart data and graphical solution data are usually output in non-readable formats to the DTF file or to other binary-data files. Summary and monitor-point/plane data are usually output in ASCII format to regular text files, possibly including the modelname.out file. User-specified data is defined and controlled via the user-subroutine facility, and can be output either to the DTF file or other binary-data files, or to regular text files.

OUTPUT CONTROL

For a steady state simulation, the user can choose from one of two options to determine when the solution data from the simulation (in graphical form) will be written to the DTF file. With the “End of Simulation” option, the solution data will be written to the DTF file only once, and this is when the maximum number of iterations has been reached or when the specified convergence criteria have been satisfied. With the “Specified Interval” option, the solution data will be written at specified intervals during the solution process, and the user has the option of creating a unique file for every output cycle, or of re-saving the solution data to the same DTF file, modelname.DTF. With the latter option, each new solution over-writes any previously saved solutions so that there is no more than one solution available at any time during the simulation. If the user chooses to have the solution data written to unique files, these unique files will be named as in the following example: modelname_steady.000025.DTF, where the number 25 in this example refers to the iteration number after which the solution has been written to that file.

For transient simulations, the user can have results written out in accordance with a specified time-step interval (that is, once every fixed number of time-steps) or in accordance with a specified integration-time interval (that is, once every fixed integration time period). With both choices, the results will be written to different DTF files numbered in accordance with the time-step number.

SUMMARY OUTPUT

Under the Summaries Section of the Out tab, the user can activate the output of the Mass Balance Summary data or the Force and Moment Summary data by clicking the corresponding check boxes. For the Mass Balance

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summaries, the data is output to a file named modelname.MASSUM. For the Force and Moment summaries, the data is output to a file named modelname.FMSUM.

The Mass Balance summary is given in the form of a tabulated list of the integrated mass flow rate through each boundary that admits flow through it. The mass flow rate is given in units of kg/s for three-dimensional domains, in units of kg/s/m for two-dimensional domains, and in units of kg/s/rad for axi-symmetric domains. The boundaries through which flow is allowed include all boundaries of the type “inlet”, “outlet”, or “fluid-fluid interface”. For output of data through boundaries of the type “fluid-fluid interface”, however, the user must make an additional election in the ACE+ GUI to specifically request output through these interfaces. If the Mass Balance Summary output is activated, the mass flow rate data for boundaries of the type “wall” or “solid-fluid interface” will also be output if there is a chemical reaction that produces or destroys mass or any other source or sink of mass at these boundaries. The data for each boundary that is included in the Mass Balance summary is given on a separate line, with the Name, the Surface ID, and the Boundary-Condition Type of the boundary given in three separate columns, and with the Inflow, Outflow, and Sum of the mass flow rate across that boundary given in another three separate columns.

The Mass Balance Summary data is also given for each grouping of boundary conditions that the user has created or defined in the ACE+ GUI, as further detailed in the User Manual. As also explained in the User Manual, for a group to be included in the Mass Balance Summary data, the group must be defined or created within the Mass Balance Summary grouping category. The data for each group included in the Mass balance Summary is given on a separate line, with the Name of the group given in the first column, and with the Inflow, Outflow, and Sum of the mass flow rate across all the members of that group given in another three separate columns.

The integral of the mass flux over a surface (which is equal to the total mass flow rate through that surface) can either be used directly as a primary solution result (for example, if a user were performing a simulation primarily to determine the mass flow rate across some portion of a device), or as an additional, secondary solution result.

The Mass Flow Balance Summary data also contains an overall summary of the net balance (or total sum) of the rates of mass generation and destruction in the computational domain and the rates of mass flow across its boundaries, and this net balance, relative to the mass inflow or outflow rate or some other appropriate scaling reference, can be used as one measure of the extent of convergence of a solution, especially for steady state, incompressible flows. By checking the “Monitor Mass Imbalance” check box, the imbalance data can be written out to a separate file MODEL_IQ.MON and viewed in CFD-ACE-PLOTTER.

The Force and Moment Summary is given in the form of a tabulated list of the pressure and viscous forces (in Newtons) integrated over each solid boundary, that is, over each boundary of the type “wall”, and “fluid-solid interface”, and the moment (in N-m) integrated over the same boundary. The moment is given in terms of its three components about the x, y and z axes, respectively. For each boundary included in the force and moment summary, the pressure forces are calculated in accordance with the following equations:

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 accordance with the following equations:


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where

μi = laminar viscosity A = face area

Vrel = relative velocity at the boundary in the x-direction, y-direction, or z-direction XN = distance from the cell center to the face center.

For each boundary included in the force and moment summary, the pressure moments are calculated in accordance with the following equations:

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 accordance with the following equations:

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.

The forces and moments on a surface can either be used directly as a primary solution result (for example, if a user were performing a simulation primarily to determine the lift or drag coefficient on a body), or as an additional, secondary solution result. The stabilization of the forces and moments on various surfaces can also be used as one measure of the extent of convergence of a solution, especially if such forces or moments were the primary solution result sought from the simulation.

BOUNDARY INTEGRAL OUTPUT

For more information on the Boundary Integral Output option, please refer to Appendix A: CFD-ACE+ Files in the ACE+ User Manual.

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DIAGNOSTICS OUTPUT

The Diagnostics data output is intended to provide the user with additional information about the execution and the status of the solver, and any problems encountered during a run. This additional information can be useful for a user interested in the execution details or statuses at various points in a simulation and also for trouble-shooting a simulation. All diagnostic data output is directed to the modelname.out file (except in the case of parallel simulations where Advanced diagnostics is requested by the user - in such cases, the diagnostic data is directed to files with names such as modelname.n.out, where n is an integer denoting the processor number outputting data to that file).

GRAPHICAL OUTPUT

Under the Graphic 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 following 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


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RESIDUAL_P Pressure Residual kg/s

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.

Variable Description Units

Mach Mach Number -



RHO Density Kg/m3

STRAIN_RATE Strain Rate* 1/s

Stream_Function Stream Function Kg/m3

U, V, W X-direction Velocity, Y-direction velocity,

Z-direction Velocity

m/s

U_absolute,

V_absolute,

W_absolute



CFL_Number CFL Number* -

WallViscousStress_X,

WallViscousStress_Y,

WallViscousStress_Z

Wall Viscous Stress N/m2

WallViscousStressMagnitude Wall Viscous Stress Magnitude N/m2

WallShearStress_X, Wall Shear Stress N/m2

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WallShearStress_Y,

WallShearStress_Z

WallShearStressMagnitude Wall Shear Stress Magnitude N/m2

SkinFrictionCoefficient Skin Friction Coefficient -

PressureCoefficient Pressure Coefficient -

Vorticity Vorticity* 1/s

Vis Effective Viscosity kg/m/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


* Additional information follows.

POST PROCESSING: ADDITIONAL INFORMATION

VORTICITY

Vorticity is calculated as follows:

STRAIN RATE

The Strain Rate in CFD-ACE+ is the magnitude of the full three-dimensional, symmetrical strain tensor. The strain rate components are shown here:

Strain Rate Components


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3-d strain tensor

Components of 3-d strain tensor

Convention for symmetric tensor

CFL NUMBER

The three choices that are available and the corresponding calculation formulae are as follows:

1. “Volume-Flow-Rate” Calculation Method: In this method, the CFL Number is computed for each cell c in the grid in accordance with the following formula:

2. "Face Velocity” Calculation Method: In this method, the CFL Number is computed for each cell face f in the grid in accordance with the following formula:

3. “Cell Velocity” Calculation Method: In this method, the CFL Number is computed for each cell c in the grid in accordance with the following formula

where in all of the above three formulae, CFLc denotes the CFL Number assigned to cell c, Δt denotes the current global time-step size, Vc denotes the volume of cell c, f in the summation operator denotes the face counter for the current cell c, nF in the summation operator denotes the total number of faces for

the current cell c, denotes the velocity vector on face f, denotes the area vector of the face f, CFLf denotes the CFL Number assigned to cell face f, Vuc denotes the volume of the upwind cell for the

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current face f, denotes the velocity vector in cell c, and n is set to the dimension of the spatial domain; that is, n is set to 2 or 3, respectively, for two-dimensional and three-dimensional domains.

For calculation-method options 1 and 3 above, wherein the CFL Number is directly calculated for each cell, the nodal values of the CFL Number are obtained by cell-to-node interpolation, and these nodal values are the values that are output by default for graphical visualization. For Option 2, wherein the CFL Number is calculated for each cell face, the face values are interpolated from the cell faces to the cell centroids, so that each cell in the grid will have an interpolated CFL Number assigned to it. These cell-centered values are then again interpolated to the nodes and output by default as nodal values.

For transient calculations, the Δt value is set to the global time-step size for the current time-step, while for steady-state simulations the Δt value is set to the arbitrary value of 1.

For definition of engineering quantities, see Appendix B.

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. The solution results 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.

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 in heat transfer problems. In CFD-ACE+, you must activate Gravity on the MO > Shared panelif 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 homogeneous fluids). Once you’ve activated it, an additional input option appears, asking you to choose how the Reference Density is calculated. The following text explains reference density and what you need to know to select the right 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 in the following figure. 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 three open boundaries and set up this type of problem with ease.


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Natural Convection on an Isothermal Heated Vertical Plate

The only drawback to this formulation is that you can no longer see hydrostatic pressure variations in CFD-VIEW; you can only see 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’ 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 ρgh, 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 we know the trivial solution to have no temperature difference and no motion. Such a problem is illustrated in the following figure, where the flow along a vertical flat plate is modeled, but the plate temperature is set to be equal to the ambient temperature.

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Test problem conditions 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 the following figure. Although the temperature field is not shown, it was checked and verified to be a constant of 292K. The resultant velocity field shown in the following figure 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.


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Solution using ideal gas law and zero pressure at boundaries

Correct Boundary Conditions For a single-phase fluid, there are at least three ways to amend the model set-up so that the model correctly reflects the physical problem being simulated:

1. Change nothing except the specification of the pressures on all inflow and outflow boundaries. In the above example, 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.

These three approaches yield the greatest simplification in the set-up of the model if the fluid has a constant density or if the fluid has a density that depends only on the temperature and not the pressure. Otherwise, these approaches will either not enable simplification of the boundary condition specifications, or have to be modified or specialized to reflect the properties of the fluid. For example, if the first approach is used with a compressible gas with large variations in altitude, then the correct exponential (as opposed to linear) variation of the boundary pressure with altitude has to be derived and specified in the boundary conditions. Similarly, if the fluid is a mixture of two phases with widely differing densities, then the third approach cannot be used on its own in a straightforward way to simplify the boundary condition specifications. Boussinesq Approximation With the Boussinesq Approximation for a single-phase homogeneous fluid, a constant fluid density is used (which is the density that corresponds to the specified reference temperature), and the buoyancy source term is calculated in accordance with the equation:

If Tref is set equal to the ambient temperature and β is set equal to 1/Tref, then there will be no externally imposed pressure gradient because there will be no source for any cell in which T = Tref.. What is being effected

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with this option is the subtraction of the hydrostatic pressure variation, which does not contribute to fluid motion, from the pressure field. If this option is used, the solution will appear as shown in the following figure. As shown in this figure, the solution has a nonzero (downward-pointing) velocity field, but the magnitude of the velocity is effectively zero.

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 = (ρ - ρref )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. The following figure 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.


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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 ρgh 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 Inertial 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 following figure shows the relax tab in the solver control panel which allows you to set under-relaxation factors for each of the solved variables and the auxiliary variables.

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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 as shown here:

Inertial relaxation 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 1e-5 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 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 0.01 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 the following figures.


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Under Relaxation for Faster Convergence

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 for troubleshooting your problem 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. 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

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corresponding solved (dependent) variable.

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

• 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. 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 such that

therefore


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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.

Examples

ESI provides users with a number of helpful tutorials. Follow these steps to locate a tutorial.

1. Go to esi-cfd.com and log on.

2. Click Model Library/Tutorials on the left side of the screen.

3. Click Search Models (sentence 2).

4. Check the box next to the model of interest, and click Submit.

The Search Results provide a list of sample models.

References

Ballyk, P.D., D.A. Steinman, and C.R. Ethier. 1994. Simulation of non-Newtonian blood flow in an end-to-side anastomosis. Biorheology. 31(5):565-86.

Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 2002. Transport phenomena. 2nd ed. New York: John Wiley & Sons, Inc. pp. 23-27, 84, 240-243, 848, 866.

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

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

Shaw, R. K. and London, A. L., Laminar Flow Forced Convection in Ducts: a source book for compact heat exchanger analytical data New York; Academic Press, 1978. (Simple Flow Theory)

Walburn, F.J. and D.J. Schneck. 1976. A Constitutive equation for whole human blood. Biorheology 13(3):201-10.

http://www.esi-cfd.com/

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WORKS CONSULTED

de Waele, A. 1923. Oil Color Chem. Assoc. J. 6:33-38.

Giersiepen M., L. J. Wurzinger, R. Opitz, and H. Reul. 1990. Estimation of shear stress-related blood damage in heart valve prostheses - in vitro comparison of 24 aortic valves. Int J Artif Organs. 13(5):300-306.

Ostwald, W. Kolloid-Zeitschrift. 1925. Berlin: Springer/Steinkopff. pp. 26, 99-117.

Veijola, T., H. Kuisma, and J. Lahdenpera. 1995. Equivalent circuit model of the squeeze gas film in a silicon accelerometer. Sensors and Actuators A 48:239-248.

Heat Transfer Module

Introduction

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. This section includes the following topics:

Applications Features Limitations Theory Implementation Examples References

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.


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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.

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

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Features The Heat Transfer Module has the following built in features:

• The ability to model ice melting problems

• The ability to model solidification problems

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

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

Feature Description

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 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.


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Heat Transfer Theory

This section describes the Energy Conservation Equation, which is the main equation that is discretized and solved in the Heat Transfer Module in the CFD-ACE+ Solver. The individual terms in this equation and how they represent different physical phenomena are explained and discussed, and the formulations and calculation procedures used to evaluate these terms when this equation is solved in conjunction with the governing equations of other modules in the CFD-ACE+ Solver are outlined. The specific numerical methods used to solve this equation are not discussed in this section but are instead detailed in the Numerical Methods section of the CFD-ACE+ User Manual.

All energy generation, consumption, conversion, or transport processes, including all heat transfer processes, are governed by the Energy Conservation Equation. The Energy Conservation Equation must, therefore, be solved to model pure energy conversion or pure heat transfer processes, and it must also be solved to model other phenomena or processes (such as chemical reaction, phase change, or compressible flow) in which the energy conversion or transport processes, even if they are not of direct interest in themselves, cannot be decoupled from the other processes of primary interest.

The Energy Conservation Equation can be expressed in many equivalent forms. The specific form adopted in the Heat Transfer Module in CFD-ACE+ Solver is the so-called “total enthalpy” form [1], which can be written in differential form as follows

(1)

where ρ denotes the density, H denotes the stagnation or total enthalpy, t denotes the time, denotes the velocity vector, keff denotes the effective thermal conductivity (including the laminar and, if applicable, also the turbulent contribution), T denotes the temperature, p denotes the pressure, denotes the viscous stress tensor (which explicitly excludes the pressure contribution), denotes the body force vector from force source or force contribution i, and denotes the energy contribution from energy source i. The total (or stagnation) enthalpy, H, is related to other thermodynamic properties via the following defining relation

(2)

in which e denotes the static internal energy, p and ρ denote, respectively, the pressure and the density, and u, v, and w denote, respectively, the components of the velocity vector in the x, y, and z coordinate directions.

The first term in equation 1 represents the rate of accumulation of total enthalpy (which includes the rate of

accumulation of total internal energy, E, where , and the rate of increase of pressure taken together as a sum). The second term in the equation reflects the convective transport of enthalpy, and this includes the net rate of outflow of total internal energy as well as the rate of work by pressure done on the surroundings (and not by the surroundings). The first term on the right-hand side of the equation represents the rate at which thermal energy is transported from the surroundings via the mechanism of conduction, as given by Fourier’s Law of Heat Transfer. The second term on the right-hand side represents the rate of increase in pressure. This term appears in the “enthalpy form” of the energy equation to cancel out the addition of the (non-conserved) pressure in the time derivative in the first term on the left-hand side of the equation. The third term on the right-hand side represents the rate of work done by the viscous stress. The fourth term on the right-hand side represents the rate at which work is done by the body forces from all such body force contributions from all modules that are active in a simulation. The fifth term on the right-hand side of the equation represents all energy sources that are not explicitly included in the other terms in the equation. It should be noted in the above that “sources” are mentioned in the generic sense, so that a source refers to both a source and a sink.

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The third term on the right-hand side of the equation, which represents the rate at which work is done by the viscous stress forces can in turn be divided into two terms, as follows:

(3)

The first term on the right-hand side of the above equation represents the rate at which “useful” work is being done by the viscous stresses. This “useful” work can result in an increase in the kinetic energy, an increase in the potential energy, or in any other (potentially reversible) form of energy. The second term on the right-hand side of the above equation represents the rate at which viscous stresses do work to deform the fluid element. Other than for fluids with elasticity or other means of mechanical energy storage from shear deformation, the work done in this manner results in the irreversible conversion of work into thermal energy, and this term is known as the viscous dissipation term.

The body forces appearing in the fourth term on the right-hand side of equation 1 may come from any number of sources, including the gravity force, electromagnetic forces, and forces from reduced models such as the fan model and the Boussinesq model, and even the required inertial forces in non-inertial references frames.

The energy sources in the fifth term on the right-hand side include all sources of energy that are generated or calculated in other modules that are active with the Heat-Transfer Module. Most notably, these sources include the energy contributions from all radiation phenomena, and this treatment is adopted here because the radiative transport equation in the CFD-ACE+ Solver is solved in separate modules that are distinct from the Heat Transfer Module. These energy sources also include sources from chemical reaction, phase change, spray-fluid interactions, and electric and magnetic phenomena that generate or absorb energy. These energy sources also include sources that are added indirectly via any momentum or mass sources.

As mentioned previously, keff in the first term on the right-hand side of equation 1 denotes the effective conductivity, which for computations involving the turbulence module includes the turbulent conductivity as well as the laminar conductivity. The specific formula used to compute keff when any turbulence model is active is as follows

(4)

where kl denotes the laminar or molecular conductivity, and where the second term denotes the turbulent conductivity, with the turbulent viscosity and specific heat capacity at constant pressure in the numerator, and the turbulent Prandtl Number in the denominator. For purely laminar flows, the second term is set to zero.

For flows with multiple species or multiple fluids, the terms ρH and denote averages over the species or the fluids involved. In particular, for multiple species, we have the relation

(5)

where ρi denotes the partial density of species i, and where Hi denotes the total enthalpy for species i, that is,

(6)

where ei is the internal energy of species i, where pi is the partial pressure of species i, where ρi is the partial density of species i, and where ui,vi, wi are the velocity components of species i in the x, y, and z coordinate


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directions, respectively, and where nSpecies is the total number of species in the mixture. The velocity for each species, includes the convective contribution as well as the diffusive contribution if the latter is non-zero, that is,

(7)

where and are the contributions to the velocity of species i from the bulk convection and from the diffusion (which includes concentration-driven and temperature-driven diffusion, that is, the diffusive velocity for species i is given by the relation

(8)

where and are, respectively, the concentration-driven and temperature-driven diffusive fluxes, as given, respectively, in equations 4-71 and 4-73.

Similarly, we have the following relation for the second term on the left-hand side of equation 1 for multi-species mixtures

(9)

where all terms are as defined previously.

For the VOF Module, where the two fluids (fluids 1 and 2) have the same pressure, temperature, and velocity, the specific form of the averaging for the terms ρH and of equation 1 takes the following forms:

(10)

where ρ1 and ρ2 denote the densities of fluids 1 and 2, respectively, where H1 and H2 denote the total enthalpies of fluids 1 and 2, respectively, and F is the volume fraction (which is by convention the ratio of the volume of fluid 2 to the sum of the volumes of fluids 1 and 2).

For the Two-Fluid (or Eulerian-Eulerian Two-Phase Flow) Module, where fluids 1 and 2 have the same pressure, but not necessarily the same temperature nor the same velocity, the specific form of the averaging for the terms ρH and of equation 1 takes the following forms

(11)

where ρ1 and ρ2 denote the densities of fluids 1 and 2, respectively, where H1 and H2 denote the total enthalpies of fluids 1 and 2, respectively, and where α1 and α2denote the phasic fractions of fluids 1 and 2, respectively.

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Implementation


This 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 Solutions - describes the Heat Module related inputs to the CFD-ACE-Solver.

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

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.

MODEL SETUP AND SOLUTION

CFD-ACE+ provides the inputs required for the Heat Transfer Module. Model setup and solution requires data for the following panels:

Problem Type Model Options Volume Conditions Boundary Conditions Initial Conditions Solver Control Output

PROBLEM TYPE

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

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.

MODEL OPTIONS

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. Model Options include the following:


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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 tab includes model options for melting ice, solidification, and moving a solid.

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 - 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).

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.

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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

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

User Subroutine (UCOND)

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:

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


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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

MIX PIECEWISE LINEAR IN T 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:

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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 Required Input(s) Polynomial Coefficients


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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 uses 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 using the Modified Eucken Model is defined as

where

ki = thermal conductivity of species i

μi = dynamic viscosity of species i

Cp,i = specific heat of species i R = gas constant

MWi = molecular weight of species i

To calculate the mixture thermal conductivity using kinetic theory, the following equation is used

where

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

ki = viscosity of species i

Φij = dimensionless quantity

and Φij is given by:

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Required Module(s) Heat, Chemistry Required Input(s) Molecular Weight of each species, Characteristic Energy, and Collision Diameter.

The latter two inputs are used to calculate Mix Kinetic Theory viscosity. These quantities must be input into 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 thermal conductivity for each species is calculated and the mixture thermal conductivity is then evaluated as follows:

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

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.


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ICE MELTING PROPERTIES

If you have activated the Ice Melting 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.

SOLIDIFICATION PROPERTIES

If you have activated the Solidification 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.

MOVING SOLID PROPERTIES If you have activated the Moving Solid feature, you can set the moving solid parameters for each solid type of volume condition.

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).

BOUNDARY CONDITIONS


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

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Walls/Rotating Walls Symmetry Interfaces Thin Walls Cyclic

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.

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) and 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 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


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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:

Heat Flux subtype panel

Option Description Constant specify a constant heat flux (W/m2) at this boundary

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

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

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

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

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

Profile from file 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 input heat flux (W/m2) as function of X, Y, Z, and T(time, s).

User Sub(ubound) heat flux (W/m2) as specified by user subroutine.

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)

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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:

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

εe = external emissivity coefficient

T∞ = temperature of the radiation source or sink


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 Solid Cell at Wall option provides a simple treatment on the heat transfer in the external solid wall (no mesh is required in the wall).

The heat transfer between the wall and fluid cell is

(2-7)

where:

Tw = wall temperature

Tc = temperature at the cell center

hf = fluid-side local heat transfer coefficient


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

(2-8)

where:

Ts = temperature of the solid cell

ks = thermal conductivity of solid

dn = thickness of the solid border

In the figure on the left, ic1 and ic2 are two cells neighboring an external wall boundary. Solid cells are added in the figure on the right..

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

For fixed flux and adiabatic wall boundary conditions:

(2-9)

With isothermal wall, Ts , 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)

where:

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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.

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.

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.

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.


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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.

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

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.

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 Heat Transfer Module-Solver Control section includes:

Spatial Differencing Scheme Solver Selection Relaxation Parameters Variable Limits Advanced Setting

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

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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 CFL based relaxation method is not available for all modules.

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:


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• 5 = Default Value • 1 = More stability, Slower convergence • 100 = Less stability, Faster convergence

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

OUTPUT OPTIONS

The output desired from the Heat Module can be specified under the Out (Output) tab in the ACE+ GUI. In terms of type and usage, the data that can be output from the Heat Module falls into five different categories: (i) restart data; (ii) graphical solution data; (iii) summary data; (iv) monitor-point or monitor-plane data; and (v) user-specified data. Restart data and graphical solution data are usually output in non-readable formats to the DTF file or to other binary-data files. Summary and monitor-point or monitor-plane data are usually output in ASCII format to regular text files, possibly including the modelname.out file. User-specified data is defined and controlled via the user-subroutine facility, and can be output either to the DTF file or other binary-data files or to regular text files.

OUTPUT CONTROL

For a steady state simulation, the user can choose from one of two options to determine when the solution data from the simulation (in graphical form) will be written to the DTF file. With the “End of Simulation” option, the solution data will be written to the DTF file only once, and that is when the maximum number of iterations has been reached or when the specified convergence criteria have been satisfied. With the “Specified Interval” option, the solution data will be written at specified intervals during the solution process, and the user has the option of creating a unique file for every output cycle or of re-saving the solution data to the same DTF file, modelname.DTF. If the user chooses the latter option, the latest solution to be output over-writes any previously saved solutions, so there will be at most one solution available throughout the simulation. If the user chooses to have the solution data written to unique files, these unique files will be named as in the following example: modelname_steady.0000025.DTF, where the number 25 in this example refers to the iteration number after which the solution has been written to that file.

For transient simulations, the user can have results written out in accordance with a specified time-step interval (that is, once every fixed number of time-steps) or in accordance with a specified integration-time interval (that is, once every fixed integration time period). With both choices, the results will be written to different DTF files numbered in accordance with the time-step number.

SUMMARY OUTPUT

Under the Summaries section of the Out tab, the user can activate the output of the Energy Flow Rate Summary by clicking the check-box labeled “Energy Balance Summary." The Energy Flow Rate Summary data is output in text format to a file named modelname.ENGSUM and the data is given there in terms of the following fields:

BOUNDARY-BY-BOUNDARY HEAT TRANSFER SUMMARY

Name The name of the boundary.

ID The Surface ID value associated with the boundary.

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Type The boundary type, such as “Wall," “Inlet," “Outlet," and “Interface.”

Area The surface area of the boundary.

COND_IN This denotes the net conduction heat flow into the computational domain (from the surroundings / exterior of the domain) through the boundary surface. If the net conduction heat flow on the boundary surface is out of the computational domain, then this quantity is set to zero. See the COND_OUT entry below for the specific way in which the COND_IN quantity is calculated.

COND_OUT This denotes the net conduction heat flow out of the computational domain (to the surroundings / exterior of the domain) through the boundary surface. If the net conduction heat flow on the boundary surface is into the computational domain, then this quantity is set to zero.

The following formula is used to calculate the values of both the COND_IN and COND_OUT quantities:

where Cond is the conduction heat transport quantity that is assigned to either COND_IN or COND_OUT for the boundary surface, kf is the conductivity on cell face f of the boundary surface, ∇ Tf is the temperature gradient across cell face f of the boundary surface, Sf is the surface area of cell face f of the boundary surface, is the outward-pointing unit normal of the cell face f of the boundary surface, and summation is carried out from cell face 1 to cell face Nbf of the boundary surface, where Nbf is the total number of cell faces in the boundary surface. It can be seen from the above formula that the quantity within the absolute value sign is the negative of the discrete integral of the heat flux over the boundary surface.

When the summation within the absolute value operator above for a particular boundary surface is found to be positive, the quantity Cond is assigned to the COND_IN variable and shown in the COND_IN column in the heat summary table, while the COND_OUT variable is set to zero. When the summation within the absolute value sign is found to be negative, the absolute value (that is, the quantity Cond as given above) is assigned to the COND_OUT variable and listed in the COND_OUT column in the heat summary while the COND_IN variable is set to zero.

Thus, in all cases, the conduction heat flow on a boundary surface is printed out as a positive value. Note that some boundary surfaces may have cell faces with a positive heat transfer and others with a negative heat transfer, but only the sum of these heat transfers is considered in determining whether a whole boundary surface is classified as a COND_IN or COND_OUT surface.

Note also that the COND_IN and COND_OUT values that are calculated as described above do not accord with the regular engineering convention in which the heat flux is calculated according to whereby heat transfer to the domain would have a negative value, while heat transfer from the domain would have a positive value. The convention adopted here is in response to user preferences for the identifications and signs of “incoming” and “outgoing” conduction heat transfers.

CONV_IN This denotes the (signed) net convectively transported heat flow into or out of the computational domain (from or to the surroundings / exterior of the domain) through the boundary surface if that boundary surface has a net mass inflow. For a boundary surface that does not have a net mass inflow, this quantity is set to zero. See the CONV_OUT entry below for


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the specific way in which the CONV_IN quantity is calculated.

CONV_OUT This denotes the (signed) net convectively transported heat flow into or out of the computational domain (from or to the surroundings / exterior of the domain) through the boundary surface if that boundary surface has a net mass outflow. For a boundary surface that does not have a net mass outflow, this quantity is set to zero.

The following formula is used to calculate the values of both the CONV_IN and CONV_OUT quantities:

where Conv is the convection heat transport quantity that is assigned to either CONV_IN or CONV_OUT for the boundary surface, ρf is the density on cell face f of the boundary surface, H0 f is the total enthalpy on cell face f of the boundary surface, is the velocity vector on cell face f of the boundary surface, Sf is the surface area of cell face f of the boundary surface, is the outward-pointing unit normal of the cell face f of the boundary surface, and summation is carried out from cell face 1 to cell face Nbf of the boundary surface, where Nbf is the total number of cell faces in the boundary surface.

It can be seen from the above formula that Conv is the negative of the discrete integral of the total enthalpy flux over the boundary surface. It should also be noted here that the actual sign of the variable Conv depends on whether the total enthalpy is positive or negative (and this is true for both mass inflow and mass outflow boundaries). In addition, the determination as to whether a boundary surface will have a CONV_IN or a CONV_OUT assignment is based entirely on the sign of the net mass flow rate across that boundary, not on the net enthalpy flow rate. It is thus possible to have inflow boundaries with both positive and negative CONV_IN values and it is possible to have outflow boundaries with both positive and negative CONV_OUT values. The situation could also be more complicated with boundaries that have inflow and outflow portions with variable values of total enthalpy.

RAD_IN This is the radiation heat flux which transports the energy into the domain through the boundary surface.

RAD_OUT This is the radiation heat flux which transports the energy out of the domain through the boundary surface.

WALL_SOURCE This denotes the sum of the user specified energy source on the boundary surface and the energy source due to surface reaction. The value of WALL_SOURCE is positive if the sources add energy to the computational domain and the value of WALL_SOURCE is negative if the sources remove energy from the computational domain.

Sum This denotes the summation of all the heat fluxes and sources over a given boundary surface. The summation is carried out using the convention that the energy added to the computational domain is positive and the energy removed from the computational domain is negative. The value of Sum is calculated using the formula given below.

Sum = COND_IN - COND_OUT +Conv + RAD_IN - RAD_OUT + WALL_SOURCE It can be seen from the above formula that Sum can be thought of as the sum of the energy transfer to the computational domain by conduction, by convection, by radiation, and by boundary sources.

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VOLUME-BY-VOLUME HEAT SOURCES SUMMARY

Following are the types of heat sources reported in this summary, as and when applicable:

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

Transient_Source This term only appears for transient cases. The transient term reports the rate of increase of the total enthalpy in the volume for the current time step. It is calculated using the formula:

Pressure_Work This is the rate of work done on the fluid per unit volume by pressure forces. It is calculated using the formula:

Dissipation_Work This term represents the rate of work done by viscous forces. It is calculated using the formula:

where S is the viscous part of the stress tensor.

Spray_Source This term represents the heat consumed or released from the spray particles or droplets during their residence in the volume (for transient cases) or the heat values for the particles or droplets in the volume (for steady cases). This includes both convective and radiative heat transfer.

Cavi_Source This term represents heat release or generation due to cavitation.

Electronic_Source This term represents the Joule heating in the volume due to the electron flow.

Magnetic_Source This term represents the Induced heating in the volume due to magnetic flow.

Plasma_Source This term includes the Joule heating due to electron flow, the Joule heating due to ion flow, and the heat release due to ion bombardment on bounding surfaces of the volume.

User_Source This term represents the energy sources and sinks applied via user defined subroutines.

Total_Source This represents the sum of all the sources and sinks in the volume.

For Radiation cases, a Radiation flux summary is printed to the modelname.out file when the Energy Flow Rate Summary is activated on the Summary sub-tab. This summary reports the radiative heat transfer for all the boundaries in the model. If the Monte Carlo radiation model is being used, then the patch type, patch temperature, and the number of rays absorbed are also reported. The following is a description of the various items reported in this summary:

Name The name of the boundary.

Surface ID The Surface ID value associated with the boundary

Type The boundary type, such as “wall”, “inlet”, “outlet”, or “interface”.

EMIT The amount of radiative heat being emitted from the boundary.


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ABSORB The amount of radiative heat being absorbed by the boundary.

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

RAD to SYS* The amount of heat contributed to the system

The Energy Flow Rate Summary written to the output file (modelname.ENGSUM) can be used to determine the bulk behavior of the solution and to judge the extent of convergence of the simulation. This is because the law of conservation of energy requires the summation of all energy transfers into and out of the computational domain to balance with the rate of accumulation of energy in the computational domain (unless heat sources or sinks are present). Therefore, the extent of imbalance in the transfer and accumulation terms gives an indication of the extent of convergence of the solution, and any solution in which the balance is not at least within one or two orders of magnitude relative to the inflow or the outflow rates of energy cannot be considered well converged. By checking the “Monitor Energy Imbalance” check box, the imbalance data can be written out to a separate file MODEL_IQ.MON and viewed in CFD-ACE-PLOTTER.

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 Unit

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 -

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.

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Post Processing Variables Variable Description Unit

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.


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.


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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:

ν = fluid velocity

μ = 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 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 • Turbulent Mixing of Propane and Air (with and without reactions) • Oil Flow through a Compliant Orifice • Multi-step Reaction in a Gas Turbine Combustor • Surface Reaction in a 2D Reactor • Generic Semiconductor Reactor

References

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

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Turbulence Module

Introduction

The Turbulence Module enables you to simulate the phenomenon or the effects of turbulence. Turbulence may have a strong influence on momentum, heat, and mass transfer. For problems with high values of the Reynolds Number, 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:

Applications Features Theory Limitations Implementation Frequently Asked Questions Examples References

Applications

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

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-ε RNG k-ε Model Realizable k-ε (RKE) Model Kato-Lauder k-ε Model Low Reynolds Number k-ε Model (Chien) Two-Layer k-ε Model k-ε Vn2F Models Wilcox 1988 (Standard) Wilcox 1988 (Corrected) Wilcox 1998 Wilcox 2006


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Wilcox 2006 LRN k-ω SST Model Spalart-Allmaras Model Large Eddy Simulations - SGS Models Smagorinsky Model Germano's Dynamic Subgrid Scale Model 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.

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.

Theory

TURBULENCE MODELS OVERVIEW

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 this section.

INTRODUCTION TO TURBULENCE THEORY

For more than a century the preferred approach in the treatment of turbulent flows has been to predict macroscopic statistics using the RANS formalism. Introduced by Reynolds in 1895, this approach involves a simple decomposition of the instantaneous fields into 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 the solution of two extra transport equations. One is for the turbulent kinetic energy, k, and the other is for the rate of dissipation, ε, or the specific rate of dissipation, ω.

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Depending on 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, ν. High Reynolds Number 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-Reynolds-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.

REYNOLDS AVERAGED NAVIER-STOKES SIMULATIONS

STANDARD K-Ε MODEL

Several versions of the k-ε 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

For the compressible turbulence flows the turbulence dissipation rate can logically be written as solenoidal dissipation (εs incompressible part) and dilation dissipation (εd). In case of incompressible flows dilation dissipation part is not important and simply neglected.

Based on Direct Numerical Simulation, Sarkar et al. (1989) and Zeman (1990) proposed that the dilation dissipation should be a function of turbulence Mach number Mt, which is defined as

Here a is the speed of sound. Turbulence dissipation rate is ε = εs + εd.

The transport equations for k and ε are,

3.3

(3-4)


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with the production term P defined as:

(3-5)

The five constants used in this model are:

In the general form εd = α1εsF(Mt), here α1 is constant and F(Mt) is compressibility function.

For the calculation of compressibility functions F(Mt) following three models are implemented in the ACE+.

1: Sarkar Correction (1989):

2: Zeman Correction (1990):

Where γ is the specific heat ration and H (x) is the Heaviside step function ψ = 0.6 and Mt0= 0.1.

3: Wilcox Correction (1992):

Here H(x) is the Heaviside step function and Mt0= 0.25.

The standard k-ε 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:

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Here yν+ 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,

(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 (σ and σt) 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.

Note The compressibility correction is only applicable for free compressible flows type of applications.

RNG K-Ε MODEL

A variation of the k-ε 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


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and ε (equation 3-13 and equation 3-14) have the same form as the standard k-ε models. The model coefficients, however, take different values as:

The coefficient Cε1 becomes a function of η, the ratio of time scales for turbulence and mean strain rate.

(3-14)

(3-15)

The constants in equation 3-14 have the values η0= 4.38 and β = 0.015. The rate of mean-strain tensor, Sij, is defined as follows:

(3-16)

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

REALIZABLE K-Ε MODEL

The realizable k-ε (RKE) turbulence model is most readily described and characterized as a variant of the standard k-ε model. In particular, the RKE model can be derived from the standard k-ε model by the following:

1. Replacing the transport equation for the rate of dissipation of turbulent kinetic energy, ε, in the standard k-ε model with a similar transport equation that models the dissipation rate according to the dynamic behavior of the mean square vorticity fluctuation in the high turbulent Reynolds Number limit; and

2. Replacing the eddy viscosity equation of the standard k-ε model with an eddy viscosity equation that ensures satisfaction of the realizability constraints (for the normal and shear turbulent stress components).

The transport equation for the turbulent kinetic energy, k, in the RKE model remains unchanged from that in the standard k-ε model, and so do many of the main modeling characteristics and parameters.

The Dissipation Rate Equations for the RKE Model

The dissipation rate equation for the RKE model is modified from the standard k-ε model to the following form:

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where

where Gb denotes the generation of turbulence kinetic energy due to buoyancy (and this term is actually omitted from the current implementation in ACE+ for compatibility and consistence with the other turbulence models implemented in ACE+), and where the parameters C2 and C1ε are constants, σε is the turbulent Prandtl number for ε, and Sε is a user-defined source term. The model constants C1ε , C2, σk and σε have been chosen to ensure that the RKE model performs well for certain canonical flows, and the specific values to which these parameters have been set in the solver are as follows:

The form of the dissipation rate, ε, equation is quite different from that of the standard k-ε model and that of the RNG model. One of the noteworthy features is that the production term in the ε equation (the second term on the right-hand side of the equation) does not involve the production of k; that is, it does not contain the same production term as the other k-ε models. It is believed that the form of the RKE model better represents the spectral energy transfer. Another desirable feature of the equation is that the destruction term (the next to last term on the right-hand side of dissipation rate equation) does not contain any singularity, that is, its denominator never vanishes, even if k vanishes or becomes smaller than zero. This characteristic contrasts markedly with other k-ε models, which have a singularity arising from the denominator being equal to k.

The Realizability Condition in the RKE Model

The term "realizable" means that the model satisfies certain mathematical constraints on the Reynolds normal and shear stress components, ensuring consistence with the physical requirements of a turbulent flow. Neither the standard k-ε model nor the RNG k-ε model is realizable.

To better understand realizability, consider combining the Boussinesq relationship and the eddy viscosity definition to obtain the following expression for the normal Reynolds stress in an incompressible, strained mean flow:

Using νt ≡ μt / ρ, one obtains the result that the normal stress, , which by definition must be a positive quantity, becomes negative; that is, "non-realizable" whenever the strain becomes large enough to satisfy

Similarly, it can also be shown that the Schwarz inequality for shear Reynolds stresses , with no summation over α and β) can be violated when the mean strain rate is sufficiently large. The most straightforward way to enforce the realizability constraints (that is, the positivity of the normal Reynolds stresses


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and the satisfaction of the Schwarz inequality for shear Reynolds stresses) is to make Cμ variable by sensitizing it to the mean flow (mean deformation) and the turbulence quantities k and ε. The notion of variable Cμ has been suggested by many modelers including Reynolds [Reynolds, 1987] and is well substantiated by experimental evidence. For example, Cμ is found to be around 0.09 in the inertial sub-layer of equilibrium boundary layers, and 0.05 in a strong homogeneous shear flow.

Modeling the Turbulent Viscosity in the RKE Model

As in other k-ε models, the eddy viscosity in the RKE model is computed from the relation

where all terms are as defined above. The difference between the realizable k-ε model and the standard k-ε and the RNG k-ε models is, as mentioned above, that Cμ is no longer constant: in the RKE model, it becomes a field variable, computed from the relation

where

and

where is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity ωk. The model constants A0 and As are given by

where

It can be seen that Cμ is a function of the mean strain and rotation rates, the angular velocity of the system rotation, and the turbulence variables k and ε. The quantity Cμ as formulated above can be shown to recover the standard value of 0.09 for an inertial sub-layer in an equilibrium boundary layer.

The Strengths, Weaknesses, and Distinguishing Characteristics of the RKE Model

The RKE model with the specific modified dissipation rate transport equation and the specific eddy viscosity model that are described above was first proposed by Shih, et al. [Shih, et al., 1996]. The aim of the new model was to overcome some of the recognized defects and deficiencies of the standard k-ε model; namely, (i) the inaccurate reflection of the turbulent length scales in the transport equation for the dissipation rate; and (ii) the over-prediction of the eddy viscosity for high shear rates. The first of these defects and deficiencies is regarded as being responsible for the so-called "round-jet anomaly”, in which the spreading rate for planar jets is predicted

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accurately but the spreading rate for axisymmetric jets is predicted poorly. The second of these defects and deficiencies is responsible for the poor performance and accuracy for flows with strong separation and recirculation.

Thus, the RKE model will predict the spreading rate of both planar and axisymmetric jets more accurately than the standard k-ε model. The RKE model (like the RNG k-ε model) will also give more accurate predictions than the standard k-ε model for flows with high mean shear rates, flows with large-scale separations, flows with strong re-circulations or strong streamline curvature, flows involving boundary layers under adverse pressure gradients, and flows with rotation. Furthermore, the improved predictions of the RKE model (relative to those of the standard k-ε model) should be in all turbulence-dependent quantities, including the wall shear stresses, the boundary-layer thicknesses, and the locations of separation and re-attachment points. Furthermore, because of the explicit enforcement of realizability, the RKE model is expected to have a slight edge over the RNG k-ε model for flows with strong separations and re-circulations.

The only known defect of the realizable k-ε model arises from the incorporation of the mean rotation in the computation of the turbulent viscosity, which may lead to non-physical turbulent viscosity predictions in simulations with multiple rotating frames or multiple reference frames. Following the explanations given above, the standard k- ε model can be expected to retain the accuracy advantage over the RKE model in flows with dominant boundary layers and in flows with weak or no separations, since by its formulation, the model dissipation equation of the RKE Model is valid in the limit of high turbulent Reynolds Number.

Other than for the above comparisons, the RKE Model retains most of the characteristics, properties, strengths, and weaknesses of the standard k-ε model (and other members of that family), especially with respect to the flow regimes for which it is most applicable, and with respect to its convergence behavior and its computational resource requirements.

KATO-LAUNDER K-ε MODEL

Another extension to the standard k-ε 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-ε Model).

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

(3-18)

where Ωij is the vorticity tensor and is defined as:

(3-19)

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


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LOW REYNOLDS NUMBER K-Ε MODEL (CHIEN)

High Reynolds Number k-ε 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-ε models that permit the integration of momentum and k-ε equations all the way to the wall. Several versions of low-Reynolds Number k-ε models have been proposed. The k-ε equations are modified to include the effect of molecular viscosity in the near wall regions. The general form of low-Reynolds Number k-ε models is given by the following equations:

(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.

TWO-LAYER K-Ε 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.

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As a compromise, 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-ε 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 ε-equation is replaced by an algebraic relation. With the two-layer model, turbulent viscosity is calculated as:

(3-24)

The damping function f μ 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 = kCμ-3/4 and the interface location is at fμ= 1, below which the rate of

dissipation is calculated as:

(3-28)

GI-LIEN V2F MODEL

The model requires the solution for k, ε, v2, and f. The first two equations are similar to the standard k-ε model.

(1)

(2)


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(3)

(4)

(5)

(6)

(7)

(8)

(9)

In addition, the velocity scale is represented by the turbulence velocity v2

(10)

where the production kf reads

(11)

with length scale L defined as

(12)

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GI-Lien version is a code-friendly formulation of V2F model.

STD-LIEN AND STD-DURBIN V2F MODEL

These versions of v2f models use the standard k-ε equations.

(13)

(14)

at the walls, ,

(15)

The transport equation for turbulence velocity is

(16)

the production term kf represents redistribution of turbulence energy. The system is closed by an elliptic relaxation equation for f

(17)

(18)

Using Boussinesq approximation for the stress-strain relation

(19)

where

(20)

Model constants are given by

(21)


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In Std-Lien version, n=6 is used, which requires the following modifications

(22)

model constraints:

(23)

Std-Lien version is more code-friendly than Std-Durbin.

LKD-LIEN AND LKD-DURBIN V2F MODEL

The first two equations for k and ε are defined as

(24)

(25)

where the time and length scales, T and L, are

(26)

Realizability constraints are use in both equations

(27)

where

(28)

Two versions of the v2 f model define the formulation for and f equations:

(29)

(30)

(31)

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(32)

n=6 corresponds to the LKD-Lien version of the model, while n=1 corresponds to the LKD-Durbin version. The model constants above are for n=1. For n=6 the following modifications for CL, Cε1, and Cη, are required.

(33)

where Rt=k2/ευ.

The LKD-Lien version is more code-friendly than the LKD-Durbin version.

WILCOX1988 STANDARD (K-Omega two equation model based on Wilcox 1st edition )

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

(3-29)

(3-30)

The transport equations for k and ω are:

(3-31)

(3-32)

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


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(3-33)

The boundary conditions for k and ω at wall boundaries are:

(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.

WILCOX1988 CORRECTED (K-Omega two equation model based on Wilcox 1st edition) (Wilcox, D.C 1993)

Transport equations for standard K-Omega model are:

Turbulent eddy viscosity is computed by

Model Constants:

WILCOX1998 (K-Omega two equation model based on Wilcox 2nd edition) (Wilcox, D.C 1998)


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Model Constants

WILCOX2006 (K-Omega two equation model based on Wilcox 3rd edition) (Wilcox, D.C 2008)



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Turbulent Production:

For incompressible flow, ∇⋅U is small and does not contribute significantly to the production term. In case of compressible flow ∇⋅U become large only in the region with higher velocity divergence.

Modulus of mean strain rate tensor:

where


where

Model Constants:

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WILCOX2006_LRN (Low Reynolds number version of k-omega two equation model ) (Wilcox, D.C 2006)



where


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Model Constants:

K-Ω SST MODEL

Menter transformed the standard k-ε model into the k-ω form, and developed a blending function F1 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-ω model is solved, and in the outer region a gradual switch to the standard k-ε 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.


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Each of the constants are blended of an inner(1) and outer(2) constant, blended by:

Here φ1 and φ2 represents respective constants. All of the additional functions are as follows:

Here d is the distance from the filed point to the nearest wall.

In ACE+, the production limiter is incorporated as follows, which is suggested in the literature:

Model Constants:


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Note Implemented k-omega and k-omega sst model are used to solve external aerodynamic type of applications

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:

The transport equation is solved using the following model constants:

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LARGE EDDY SIMULATIONS

INTRODUCTION TO LARGE EDDY SIMULATIONS

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 and Δ f 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.

SGS MODELS

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.

SMAGORINSKY MODEL

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)


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

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 Δ 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.

MENON'S LOCALIZED DYNAMIC SUBGRID-SCALE MODEL

The Localized Dynamic Subgrid-Scale model (LDKM) uses scale-similarity and the subgrid-scale kinetic energy:

(3-40)

to model the unresolved scales. Using kSGS the SGS stress tensor is modeled 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:

(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, εSGS based on simple scaling arguments:

(3-45)

In these models, Cτ and Cε 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

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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:

(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 Cε 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.


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DETACHED EDDY SIMULATION (DES)

Detached-Eddy Simulation (DES) is a hybrid technique for prediction of turbulent flows at high Reynolds numbers. Development of the technique was motivated by estimates which indicate that the computational costs of applying Large-Eddy Simulation (LES) to complete configurations such as an airplane, submarine, or road vehicle are prohibitive. The high cost of LES when applied to complete configurations at high Reynolds numbers arises because of the resolution required in the boundary layers, an issue that remains even with fully successful wall-layer modeling.

In Detached-Eddy Simulation (DES), the aim is to combine the most favorable aspects of the two techniques, i.e., application of RANS models for predicting the attached boundary layers and LES for resolution of time-dependent, three-dimensional large eddies. The cost scaling of the method is then favorable since LES is not applied to resolution of the relatively smaller-structures that populate the boundary layer.

THEORY

The base model employed in the majority of DES applications to date is the Spalart-Allmaras one-equation model (Spalart and Allmaras 1994, referred to as S-A throughout). The S-A model contains a destruction term for its eddy viscosity which is proportional to , where d is the distance to the wall. When balanced with the production term, this term adjusts the eddy viscosity to scale with the local deformation rate S and d: . Subgrid-scale (SGS) eddy viscosities scale with S and the grid spacing Δ, i.e., . A subgrid-scale model within the S-A formulation can then be obtained by replacing d with a length scale directly proportional to the grid spacing.

To obtain the model used in the DES formulation, the length scale of the S-A destruction term is modified to be the minimum of the distance to the closest wall and a lengthscale proportional to the local grid spacing, i.e.,

. In RANS predictions of high Reynolds number flows the wall-parallel (streamwise and spanwise) spacings are usually on the order of the boundary layer thickness and larger than the wall-normal spacing. Choosing the lengthscale Δ for DES based on the largest local grid spacing (i.e., one of the wall-parallel directions) then ensures that RANS treatment is retained within the boundary layer, i.e., near solid walls, d << Δ and the model acts as S-A, while away from walls where Δ << d, a subgrid model, is obtained.

Basic equation - The DES formulation in this study is based on a modification to the S-A model such that it reduces to RANS close to solid surfaces and to LES away from the wall (Spalart et al. 1997). The S-A RANS model is written as (see Spalart and Allmaras 1994),

(3-53)

Where is the working variable. The eddy viscosity υt is obtained from

(3-54)

Where υ is the molecular viscosity. The production term is expressed as

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(3-55)

(3-56)

where S is the magnitude of the vorticity. The production term as written in (3) differs from that developed in Spalart and Allmaras (1994) via the introduction of and re-definition of . These changes do not alter predictions of fully turbulent flows and have the advantage that for simulation of flows with laminar separation, numerical diffusion upstream of the eddy viscosity into attached, laminar regions is prevented. The function is given by

(3-57)

The function ft2 is defined as

(3-58)

The trip function ft1 is specified in terms of the distance dt from the field point to the trip, the wall vorticity ωt at the trip, and ΔT which is the difference between the velocity at the field point and that at the trip,

(3-59)

where gt = min (0.1, ΔU/ω, Δχ) and Δχ is the grid spacing along the wall at the trip. For the current simulations, ft1 was set to zero to alleviate the high cost of evaluating dt. For simulations in which the flow is tripped, large levels of eddy viscosity are added at designated trip locations. The wall boundary condition is . The constants are:

The DES formulation is obtained by replacing the distance to the nearest wall, d by , where is defined as

(3-60)

In Eqn. (3-60) for the current study, Δ is the largest distance between the cell center under consideration and the cell center of the nearest neighbors (i.e., those cells sharing a face with the cell in question). In "natural" applications of DES, the wall-parallel grid spacings (e.g., streamwise and spanwise) are on the order of the boundary layer thickness and the S-A RANS model is retained throughout the boundary layer, i.e. = d. Consequently, prediction of boundary layer separation is determined in the 'RANS mode' of DES. Away from solid


128

boundaries, the closure is a one-equation model for the SGS eddy viscosity. When the production and destruction terms of the model are balanced, the length scale in the LES region yields a Smagorinsky eddy viscosity

. Analogous to classical LES, the role of Δ is to allow the energy cascade down to the grid size; roughly, it makes the pseudo-Kolmogorov length scale, based on the eddy viscosity, proportional to the grid spacing. The additional model constant was set in homogeneous turbulence.

Descritizations - FVS, FDS and AUSM (explicit only) can be employed to descritize the above scalar transport equations, which are default options for general use in FASTRAN GUI.

Boundary conditions - In addition to normal boundary conditions used in other turbulence models, DES also accepts the random inlet boundary conditions for velocities. Two types of random inlets are supported: Gaussian random and time-correlated Gaussian random. Both the Gaussian and time correlated methods randomly perturb the mean inlet velocity components over the faces of the inlet boundary using a Gaussian profile of the root mean squared (RMS) turbulent intensities at each time step. The time correlated method further specifies that the perturbation is correlated over a length of running time. In Gaussian random method, Box-Muller transform is employed to generate random number and is a method of generating pairs of independent standard normally distributed random numbers, given a source of uniformly distributed random numbers.

It is commonly expressed in two forms. The basic form as given by Box and Muller takes two samples from the uniform distribution on the interval (0, 1] and maps them to two normally distributed samples. The polar form takes two samples from a different interval, [−1, +1], and maps them to two normally distributed samples without the use of sine or cosine functions.

Properties - No specific requirement.

USER INPUT

Activation of this model is under MO > Flow tab as shown Fig. 1. Typical value of the DES constant is 0.65, but FASTRAN also allows users to change as they like. The variation of DES constant is just for tuning specific cases, which would be benefit and convenience to and for user’ research orientation. It is strongly recommended that user should keep the default value for DES simulations. Random inlet boundary conditions can be specified for DES model as shown in Fig. 2.

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Figure 1: Activation of DES model

USER OUTPUT

The output for DES is the same as that for S-A turbulence model.

DES LIMITATIONS

Like LES model, physically DES is only applicable for transient flow problems. One can carry out steady simulations with DES, but the results do not serve as physical meanings. At present, DES model is only available for structured grid solver in FASTRAN. Also it is recommended that user employ DES for accurate predictions involving massive separation flows.

DES FUTURE EXTENSIONS

While DES has demonstrated that it is relatively well understood in massively separated flows characterized by thin boundary layers prior to separation, an incorrect behavior can be encountered in flows with thick boundary layers and/or shallow separations. To address these deficiencies, new modified DES known as Delayed-Detached Eddy Simulation (DDES) has been developed. For next release, DDES turbulence model will be implemented into FASTRAN.


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Figure 2: Random inlet boundary conditions specifications

DES TEST CASES

Two cases have been performed to test the implementation of DES turbulence model; one is 2D NACA0012 airfoil, the other 3D box. For both cases, the DES constant CDES is equal to 0.65.

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NACA0012 airfoil DES implemented was firstly used to compute the flow over a NACA0012 airfoil at a Reynolds number of 6*104 and a Mach number of 0.00288 with angle of attack = 0. The configuration, computing conditions and the results are shown in Fig. 3 (a-f).

3D box DES is then applied to predict the flow around a 3D box at a Reynolds number of 3*104 and a Mach number of 0.00288 with angle of attack = 0 . It has been shown that over a wide range of Reynolds numbers the flow is characterized by the vortex shedding with large-scale vorticity emanating from the shear layer which separates from surface of the box. The main goal of this case is to compare the results obtained using traditional RANS and DES. The comparison includes the time history of integral parameters such as drag coefficient and mean distribution of pressure around the box. The configuration is shown in Fig. 4, and computing conditions and the results are shown in Fig. 5 (a-e).

SPALART REFERENCES

Spalart, P.R., Allmaras, S.R.: A one-equation turbulence model for aerodynamic flows, La Recherche Aerospatiale, 1 pp. 5-21, (1994).

Spalart P.R., Jou W.H., Strelets M., Allmaras S.R.: Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach, 1st AFOSR Int. Conf. on DNS/LES, Aug. 4-8, 1997, Ruston, LA. In: Advances in DNS/LES, C. Liu and Z. Liu Eds., Greyden Press, Columbus, OH, USA (1997).

(a) t = 0.1 s

(b) t = 1 s


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(c) t = 2 s

(d) t = 3 s

(e) t = 4 s

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(f) t = 5 s

Fig. 3 Comparison between DES (LEFT) results and S-A (RIGHT) results for NACA0012 airfoil

Fig. 4 A 3D box geometry


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(a) t = 0.1 s

(b) t = 0.5 s

(c) t = 1 s

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(d) = 1.5 s

(e) t = 2 s

Fig. 5 Comparison between DES (LEFT) results and S-A (RIGHT) results for a 3D box

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).

TRANSPORT EQUATION FOR INTERMITTENCY

The intermittency equation is formulated as follows:


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(1)

The source terms are defined as

(2)

(3)

where S is the strain rate magnitude. Fonset, 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, Reθt in equation 5, is related to the transition momentum thickness Reynolds number according to

(10)

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

The destruction terms are defined as

(11)

(12)

where Ω 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-laminarization. The function

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Fturb 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 γ at a wall is zero normal flux while at inlet is equal to 1. For accurate transition the grid must satisfy y+≈ 1. If y+ is too large (5), the transition onset location moves upstream with increasing y+. 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 Flength and at present are proprietary to CFX. The following correlations have been proposed to replace the proprietary versions,

(17)

(18)

TRANSPORT EQUATION FOR TRANSITION MOMENTUM THICKNESS REYNOLDS NUMBER

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

(1)

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

(2)

(3)

with being a time scale for dimensional reason. The blending function Fθt 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


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(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.

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

(1)

(2)

(3)

(4)

COUPLING WITH TURBULENCE MODEL Although in principle the intermittency function obtained from the γ-Reθt 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

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(3)

(4)

with Pk and Dk the production and destruction terms from the original SST turbulence model and γeff obtained from above. The blending function F1 is responsible for switching between the k-ω and k-ε models is modified as follows

(5)

(6)

(7)

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

EMPIRICAL CORRELATIONS

The quantity Reθt in equation 2 of the Transport Equation for Transition Momentum Thickness Reynolds Number chapter has been determined from a correlation Reθt= F(Tu, dU/ds) where Tu is the local turbulence intensity and dU/ds 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.

ABU-GHANNAM AND SHAW CORRELATION

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

(1)

(2)

(3)

where:

(4)

MENTER CORRELATION

The empirical correlation of Menter is defined as follows:


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(1)

(2)

(3)

where:

(4)

(5)

(6)

(7)

(8)

(9)

(10)

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

(11)

(12)

(13)

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Implementation

OVERVIEW 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

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

GRID GENERATION OVERVIEW 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-w 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-ε turbulence model and output the R grid parameter:

where:

le = the energetic scales (k2/3 / e)

ld = the dissipation scales (v3 / e)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.

KOLMOGOROV MICROSCALES The parameters Kolmogorov length scale (K_length), Kolmogorov time scale (K_time) and Kolmogorov velocity scale (K_velocity) are indicative of the smallest eddies present in the flow.


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Numerically these quantities are calculated as follows:

K_length =

K_time =

K_velocity =

Where υ is Kinematic Viscosity, and ε is Turbulent Dissipation rate. These variables are obtained in CFD-VIEW on activation of grid parameter "R" in graphical output of CFD-ACE+.


MODEL SETUP AND SOLUTION OVERVIEW

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


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

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.

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Model Options Panel - Turbulence Tab

Select a turbulence model from the pull-down menu. The options are described in the Turbulence-Theory section and include:

Reynolds Averaged Navier-Stokes Simulations

k-ε Family

Standard k-ε Model

RNG k-ε Model

Realizable k-ε (RKE) Model

Kato-Launder k-ε Model

Low Reynolds Number k-ε Model (Chien)

Two-Layer k-ε Model

k-ε ν2F Models

k-ω Family

Wilcox 1988 (Standard)

Wilcox 1988 (Corrected)

Wilcox 1998

Wilcox 2006

Wilcox 2006 LRN

k-ω SST Model

Spalart-Allmaras Model

Large Eddy Simulations

Smagorinsky Model

Germano's Dynamic Subgrid Scale Model

Menon's Localized Dynamic Subgrid Scale Model

Constant Turbulent Viscosity

User Defined Turbulent Viscosity


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Turbulence in Porous Media

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.

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.

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 st 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.

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)

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VOLUME CONDITIONS

No volume condition inputs are required for the Turbulence Module.

BOUNDARY CONDITIONS

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:

Inlets Outlets Turbulent Kinetic Energy Random Inlets Walls Rotating Walls Symmetry Interfaces Thin Walls Cyclic

INLETS

The Turbulence Module needs to know how to set the turbulence quantities at inlet boundaries. The turbulence quantities that need to be specified for the turbulence models are:

Turbulence Quantity Model

Turbulent Kinetic Energy (K) k-ε or k-ω Models Turbulent Dissipation Rate (D) k-ε or k-ω Models Eddy Viscosity (Nu(t)) Spalart-Allmaras Model SGS Turbulent Kinetic Energy (Ksgs) localized dynamic SGS

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).

TURBULENT KINETIC ENERGY Turbulent kinetic energy can be specified as:

• Constant


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• 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 QUANTITIES USING INTENSITY The turbulence intensity, I, is defined as the ration of the root-mean-square of the fluctuation velocity, u', to the

mean flow velocity, .

The turbulence intensity generally ranges from 1% to 10%. That with the 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 where Re is the Reynolds number.

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 LENGTH, SCALE, AND HYDRAULIC DIAMETER

The turbulence length scale, l, 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, l is restricted by the size of the duct. A relationship between l and the hydraulic diameter L is l=0.03L.

Guidelines for choosing hydraulic diameter L or turbulence scale l:

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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 layers, choose the turbulence length scale. Set l=0.4 δ. is the thickness of the boundary layer.

TURBULENT DISSIPATION RATE For boundaries and volumes (initialization) with turbulence length scale or hydraulic diameter as input option, the dissipation rate, ε (or the specific rate of dissipation, ω) can be calculated as:

k-ε model

k-ω model

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.


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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.

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.

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FORMULATION

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)

where is the roughness height and E0 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 E to in the existing wall-function method. In so doing, is evaluated through equation 2, and it is set to 30/9 once it falls below this value for reasons discussed above.

ROTATING WALLS

Rotating walls, just like plain walls, can have a roughness height (RH) value assigned.

SYMMETRY

The symmetry boundary condition is a zero-gradient condition. There are no Turbulence Module related values for symmetry boundary conditions.

INTERFACES

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.


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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.

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.

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, 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.

SOLVER CONTROL

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

Output for LES Spatial Differencing Scheme Solver Selection

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Under Relaxation Parameters Variable Limits Advanced Settings

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.


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.

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.

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. 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


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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.

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).

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.

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

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• 1 = More stability, Slower convergence • 100 = Less stability, Faster convergence

OUTPUT

Click the Out tab to see the Output settings in the Control Panel. The output section includes:

Summary Output Graphical Output LES Output

SUMMARY OUTPUT

There are no summary outputs available for the Turbulence Module.

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 Eddy Viscosity m2/s Effective Viscosity (sum of turbulent and laminar viscosity)

kg/m-s

Y+ (only output at walls) -

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


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Ctau - LD Ceps - LD SGS Kinetic Energy m2/s2 LD SGS Dissipation Rate m2/s3 LD Test Filter Kinetic Energy m2/s2 LD

m/s S,D,LD

* S - Smagorinsky D - Dynamic LD - Localized Dynamic

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 are good indications 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-w model) Specific rate of dissipation (k-w 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 - Ywall * Distance to nearest wall m

* Ywall is only computed for the following four models: K Epsilon-Two Layer, K Epsilon - Low Re (Chien), SST K

Omega, and the Spalart-Allmaras.


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-ε model can be chosen for most problems. If there is separation or strong recirculation (such as with flow around a bluff body), the RNG or RKE models are better choices. For flows with strong stagnation the Kato-Launder model becomes the better choice. On the other

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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 activate the RKE Model and what are the usage guidelines for this model

If the RKE Model is chosen, then by its formulation, it must always be used with appropriate wall functions, such that the y+ value is no less than 11.5, and preferably closer to 30, 60, or 80, and with the same guidelines and limitations that apply for any High Reynolds Number k-ε model. The RKE model can be activated from the GUI as a separate new turbulence model, and the chosen wall function can also be selected from the GUI just as it would be for all other turbulence models that require a wall function. The wall functions that can be chosen for the RKE model are currently the following: (i) the Standard Wall Function; (ii) the Two-Layer Wall Function; and (iii) the Non-Equilibrium Model. If the RKE model is activated in conjunction with the multiple reference frame (MRF) capability, then an additional switch button appears to enable the user to switch the MRF either "on" or "off" for each wall patch, to either turn on the MRF capability for the RKE model or to turn it off, respectively, for that wall patch (with "on" signifying rotation, and "off" signifying no rotation). The RKE Model runs in parallel mode, and with all the other modules that are currently coupled with the turbulence module.

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 e for which the calculated turbulent 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.

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:


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• Tutorial 7, Turbulent Mixing of Propane and Air in Tutorial Manual, Volume II, (with and without reactions).

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.

Bredberg, J., (2000), On the Wall Boundary Condition for Turbulence Models, Chalmers University of Technology Goteborg, Sweden

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-ε Predictions of Heat Transfer in Turbulent Recirculating Flows Using an Improved Wall Treatment.” Numer. Heat Transfer 15(1989): 21-47.

Craft T. J., Gerasimov, A. V., Iacovides H., Launder B. E., (2002), “Progress in the generalization of wall-function treatments“, Int. J. Heat and Fluid Flow 23, 148-160.

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.

Guézengar, D., Francescatto, J., Guillard, H., Dussauge, J.-P. 1999, Variations on a k–epsilon turbulence model for supersonic boundary layer computations, Eur. J. Mech.B/Fluid 18 713-738.

Gutmark E and Wygnanski I. The planar turbulent jet, Journal of Fluid Mechanics, 73(3), 465-495, 1976.

Hellsten, A., "Extension of the k-ω-SST turbulence model for flows over rough surface.” AIAA-97-3577.

Kim, S.-E. and Choudhury, D., (1995) A near-wall treatment using wall functions sensitized to pressure gradient. ASME FED Separated and Complex Flows. ASME, 217, 273 –279.

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. (1974):3 269-289.

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-ω turbulence models for aerodynamic flows." AIAA-93-2906.

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Menter, F. R., "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications," AIAA Journal, Vol. 32, No. 8, August 1994, pp. 1598-1605.

NASA Langley Research Center. Turbulence Modeling Resource [Internet]. Last Updated: 29 August 2012. http://turbmodels.larc.nasa.gov/wilcox.html.

Pope, Stephen B. Turbulent Flows, Cambridge UK. Cambridge University Press, Aug 10, 2000 - 806 p.

Sarkar S. and Balakrishnan L., 1990, Application of a Reynolds stress turbulence model to the compressible shear layer, NASA Report-ADA-227097.

Sarkar S., Erlebacher G., Hussaini M.Y., Kreiss H.O., The analysis and modelling of dilatational terms in compressible turbulence, ICASE Report 89-1789, 1989.

Shih, T.-H. , Liou, W. W. ,Shabbir, A. , Yang, Z. and Zhu, J. A New k-e Eddy-Viscosity Model for High Reynolds Number Turbulent Flows - Model Development and Validation, Computers and Fluids, 24(3): 227-238, 1995.

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. Turbulent Mixing of Propane and Air, CFD-ACE Tutorial, ESI-CFD, Inc, AL, 2009.

Tennekes, H. and J. L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, MA (1972)

W. C. Reynolds. Fundamentals of turbulence for turbulence modeling and simulation, Lecture Notes for Von Karman Institute Agard Report Number 755, 1987.

Wilcox D.C., 1992, Dilatation–dissipation corrections for advanced turbulence models, AIAA J. 30 (1992) 2639–2646.

Wilcox, D. C., "Formulation of the k-omega Turbulence Model Revisited," AIAA Journal, Vol. 46, No. 11, 2008, pp. 2823-2838

Wilcox, D. C., Turbulence Modeling for CFD, 1st edition, DCW Industries, Inc., La Canada CA, 1993.

Wilcox, D. C., Turbulence Modeling for CFD, 2nd edition, DCW Industries, Inc., La Canada CA, 1998.

Wilcox, D. C., Turbulence Modeling for CFD, 3rd edition, DCW Industries, Inc., La Canada CA, 2006.

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.

Zeman O., Dilatation dissipation: the concept and application in modeling compressible mixing layers, Phys. Fluids A-Fluid 2 (1990) 178–188.


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Chemistry

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.)

This sectiontopic includes the following subjects:

Applications Features Limitations Theory Implementation Frequently Asked Questions References

Applications

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 do the following:

• Study processes such as deposition and etching that are vital in semiconductor processing applications. • Run mixing-only cases and gas-phase and/or surface reactions prescribed within the volumes and/or at

surfaces. • Work with other CFD-ACE+ modules to study multi-physics problems.

Topics in this section include the following:

Mixing Only Mixing with Gas Phase Reactions Mixing with Surface Reactions Multi-Physics Applications

MIXING ONLY

Use the Chemistry Module to simulate mixing two or more inert species or mixtures, and to obtain the spatial and temporal variation of the species concentrations.

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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.

MIXING WITH SURFACE REACTIONS

Use the Chemistry Module to model surface reactions occurring in chemical vapor deposition (CVD) systems.

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

Features

The Chemistry Module has many inherent features which may or may not be activated for any given simulation. This section includes the following topics:

Solution Approach Mass Diffusion Options Gas Phase Reactions Surface Reactions Coupled Solver Unsteady Combustion

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.


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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

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.

GAS PHASE REACTIONS

The Chemistry Module contains the following gas phase reaction models:

Instantaneous Reaction Model (for Mixture Mass Fraction approach) Equilibrium Reaction Model (for Mixture Mass Fraction approach) Finite-Rate Model (for Mixture Mass Fraction approach) Finite-Rate Model (for Species Mass Fraction approach)

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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.

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.

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.

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.

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 reaction, the backward reaction rate can be computed using prescribed values, or by using equilibrium.

EDDY BREAKUP MODEL

You can use the eddy breakup model for turbulence-combustion interaction for turbulent flows, with any of the k-e 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.


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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.

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.

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.

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 the case of complex reaction mechanisms, the Laminar Chemistry Operator Splitting option allows you to use 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 sub-grid chemistry closure (the Linear Eddy Model) is available along with the ISAT and Staggered Chemistry options.

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Limitations

The Chemistry Module does include a few limitations:

• The diffusion coefficient (G) 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 diffusivity of all species is 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.

Theory

INTRODUCTION

The Chemistry Module enables you to model mixing and reacting flow systems. This Theory section discusses the following topics:

Basic Definitions And Relations Gas Phase Reaction Models Surface Reaction Models Combustion Interaction

DEFINITIONS AND RELATIONS

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 φ CO2 + 2 H2O (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 x CH3+ H2 (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:


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Composition Variables Chemical Rate Expressions Third Body Reactions Mixture Fractions

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.

(3)

The mixture molecular weight is given by:

(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:

(5)

where ρ, 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:

(6)

and is a useful quantity in converting concentration units, as can be seen by examining equations 3 through 5. The partial pressure of species i in a mixture of gases is defined as:

(7)

CHEMICAL RATE EXPRESSIONS

A system of Nrxn chemical reactions involving Nsp species can be expressed in a general notation by

(8)

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where Λi is the chemical symbol for species i, ν'ij and ν''ij are the forward and reverse stoichiometric coefficients for the ith species in the jth reaction. Equation 8 can be written more compactly as:

(9)

where νij= ν''i -ν'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 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:

∧i v'i1 v'i2 v"i1 v"i2 H2 0 2 1 0 H2O 1 0 0 2 CO 1 0 0 0 CO2 0 0 1 0 O2 0 1 0 0 N2 0 0 0 0

The molar production rate of species i due to chemical reaction is

(10)

The rate-of-progress variable for the jth reaction, qj, can be generally expressed as:

(11)

where (kf)j and (kr)j are temperature-dependent forward and reverse rate coefficients, and and are constants.

For elementary reactions which obey the mass action law, = and = where and are the stoichiometric coefficients defined in equation 8.


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The concentration exponents in equation 11 are not necessarily related to the stoichiometric coefficients for global reactions.

The rate coefficients are assumed to have an Arrhenius form:

(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 11 are (moles/volume/time). The units of A, therefore, depend on the exponents of the molar concentrations in equation 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

(13)

the rate of the reaction is expressed as

(14)

where is expressed in kmoles/m3s.

The units on Ap are dependent upon α, β, and n as shown below:

(15)

The reverse rate coefficient can be obtained from the equilibrium constant, Kc , for reactions obeying the law of mass action:

(16)

The equilibrium constant (actually a function of temperature) can be calculated from thermodynamic data:

(17)

where:

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p0 = reference pressure of one atmosphere

= Gibbs free energy of species i at one atmosphere.

THIRD BODY REACTIONS

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:

(18)

\where γij is third body efficiency and βij is order for third-body.

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

(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 (Γ) 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 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 19 over all mixtures. Let xik denote the mass fraction of the ith species in the kth mixture. It is easily shown that when equation 19 is multiplied by xik for each mixture fraction and summed over all mixture fractions, the following equation is obtained.


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(20)

where:

(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 21 when the diffusion coefficients of all species are equal. The boundary conditions for the mixture fractions are defined such that the boundary conditions for the mass fractions are satisfied by equation 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.

GAS PHASE REACTION MODELS

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)

INSTANTANEOUS CHEMISTRY MODEL

In the instantaneous chemistry model, the reactants (species on the left-hand side of equation 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 21 to calculate the composition that would occur without the reaction. The unreacted composition, denoted by the superscript "u”, is given by

(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 diffusivity of all species are equal.

A stoichiometrically correct reaction step needs to be specified. Consider a single reaction between λ1 (fuel) and λ2 (oxidizer) to produce an arbitrary number of product species.

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(23)

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:

(24)

The stoichiometric coefficients in equation 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:

(25)

The change in each species due to the reaction is proportional to the change in fuel, with the proportionality constant given by equation 24. The mass fraction of each species is then given by:

(26)

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.

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:

(27)

where:

μi = 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.


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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:

(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:

(29)

The quantities λj are termed Lagrangian multipliers. Because equation 28 must be satisfied to conserve elements, the second term on the right side vanishes and the composition that minimizes Ψ also minimizes G. Differentiating equation 29 with respect to ni gives:

(30)

Differentiating equation29 with respect to λi gives:

(31)

Setting equations 30 and 31equal to zero gives N + M equations to be solved to give the composition at chemical equilibrium. With some algebraic rearrangement, this yields the following:

(32)

(33)

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Equation 32 is simplified as shown below:

(34)

Substituting equation34 into equation 33 yield the following:

(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 21, giving:

(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 103. 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.

FINITE-RATE MODEL (FOR MIXTURE SOLUTION)

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 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:

(37)

An Arrhenius form ( equation 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

(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.


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(39)

where ΔY1 =( Y1)u - Y1 and ri is given by equation 93. The only difference between equation 39 for the finite-rate chemistry model and equation 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.

FINITE-RATE MODEL (FOR SPECIES SOLUTION)

This finite-rate model allows for specification of single or multiple reaction steps (see equation 8) to model the process. This multi-step mechanism can be generally represented as:

(40)

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:

(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 diffusivity of individual species do not have to be equal with this chemistry model. The production rate of species i,ωi , is given by equation 10.

The source term is linearized to improve convergence.

(42)

where n and n+1 denote the iteration at which the corresponding quantity is evaluated. There are two methods available for the solution of equation 41. The first uses the full Jacobian array in equation 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.

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:

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(43)

where:

aij = gas species stoichiometric coefficient

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:

(44)

where kfj represents forward rates and krj represents 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:

(45)

and the surface concentrations are expressed as

(46)

where:

ρw = gas -phase mass density in kg/m3

ρs = surface site density in kmol/m2

= gas-phase mass fractions adjacent to the wall

Xi = surface site fractions


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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

(47)

(48)

where, the left-hand side of equation 47 is the diffusive flux of species i normal to the surface and the right-hand side of equation 47 is the production rate of species i per unit area of surface, on a mass basis. Equations 47 and 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 44) to evaluate the reaction rate.

For sticking coefficient expression, surface reaction rate equation 44 becomes:

(49)

where sticking probability (The probability that a molecule will adsorb upon collision with the reacting surface 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:

(50)

To fit into the format of equation 44, the above rate can be expressed as:

(51)

where:

(52)

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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, eki, mki, and xki are the three coverage parameters, and Xkn is the surface site fraction of the kth surface species on site n.

COMBUSTION INTERACTION

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.

(53)

where is the joint PDF of the N mass fractions at the position x and time t, and is an arbitrary function of the mass fractions.

Favre-averaged quantities can be calculated by defining a Favre-averaged PDF:

(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 diffusivity of all species must be equal and no more than two mixtures can be defined.

The Turbulence Combustion Interaction section includes the following topics:

Determining PDF


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Determining Averaged Variables Operator Splitting In Situ Adaptive Tabulation (ISAT) Subgrid Linear Eddy Model Application to Large Eddy Simulation

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:

(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.

(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 19 and equation 38. The source term due to chemistry in equation 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.

(57)

(58)

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.

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(60)

The parameters for the top-hat PDF are given below, as functions of the average and variance of the mixture fraction.

(60-a)

(60-b)

(60-c)


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(60-d)

The beta PDF is a continuous distribution defined between the values of 0 and 1.

(61)

The parameters for the beta PDF are:

(62)

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.

(63)

(64)

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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:

(65)

where c1, c2, and c3 are the probabilities of the reaction progress equaling , 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.

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 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.

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 Δt, 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:


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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 ε = |BGδφ|

where δφ is the displacement from the originating point and B is a scaling matrix.

In order to have a valid linear approximation, the error ε should be less than the specified tolerance εtol which, in combination with the intrinsic properties of the mapping gradient matrix G, defines a region of accuracy for each tabulation point φ0.

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 jq 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 Δt, and a given tolerance, the 2014-01-06 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 jq, the binary tree nodes are used to select the leaf that is likely to be the closest to jq, by determining the position of query point with respect to each cutting plane. If jq 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 the following figure). Otherwise, the

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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 f0 and jq. The entries in the tree node are the parameters of the cutting plane between the two new children.

Illustration of the EOA Growth Process

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,

(66)

where f 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,

(67)


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where

= 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 66 and equation 67.

The influence of turbulent stirring is modeled stochastically 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: l, 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 l and f(l). For a high Reynolds number turbulent flow described by a Kolmogorov cascade, the result is (McMurtry, Menon, and Kerstein, 1992):

(68)

(69)

where ReL is the Reynolds number based on the integral length scale, n is the kinematic viscosity, η 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.

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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 66 and equation 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.

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 66 and equation 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.

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 equations 68 and 69), molecular diffusion ( equation 66), and chemical reaction ( equation 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


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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.

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.

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Implementation


The Implementation section describes how to set up a model for simulation using the Chemistry module. The Chemistry 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). The general grid generation concerns apply; that is, ensuring that the grid density is sufficient to resolve solution gradients, minimizing skewness in the grid system, and locating computational boundaries in areas where boundary values are well known.

See the CFD-ACE+ User Manual > Control Panel for details on entering data using any of the panels described in this section:


PROBLEM TYPE

Click the Problem Type [PT] tab to open the Modules panel. Select Chemistry to activate the Chemistry module. This 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.

Note Do not activate the Chemistry module with the following modules: Cavitation, Free Surface, or Two Fluid Modules.

MODEL OPTIONS

Click the Model Options [MO] tab to open the Model Options panel. This section discusses the following MO options:

Shared Tab Chem Tab

Chemistry Media Gas Phase

Special Discussions Unsteady Combustion In Situ Adaptive Tabulation (ISAT)


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SHARED TAB

There are no settings under the Shared tab that affect the Chemistry module.

CHEM TAB

The model options for the Chemistry module are located on the Chem panel. The Chemistry Media > Media field contains two choices: Gas Phase and Liquid Phase. The Gas Phase > Solve For field contains choices for Mixture Mass Fractions and Species Mass Fractions. You can also select Gas Phase Reaction.

Model Options Panel

Chemistry Media, Media Gas Phase

Use this option to study gas-related problems (Mixture Mass Fractions and Species Mass Fractions).

Liquid Phase

Use this option to study electrochemistry problems (Biochemistry and General Liquid Chemistry).

Gas Phase, Solve For Species Mass Fractions

Uses a Finite-Rate (for Species Fraction Approach) mechanism.

Mixture Mass Fractions

Applies any reaction mechanism that is Instantaneous, Equilibrium, or Finite-Rate.

Equilibrium Products CO Oxidation Step Thermal NOX Nitrous NOX Prompt NOX

Gas Phase, Gas Phase Reaction

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MEDIA GAS PHASE

When you select Chem Tab > Chemistry Media > Media > Gas Phase, the gas phase section of the MO panel opens. It enables you to 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 CFD-ACE+ User Guide > Database Manager).

The Solve For field contains two choices: Mixture Mass Fractions and Species Mass Fractions (for gas phase reactions).

Solve for Mixture Mass Fractions

Use Mixture Mass Fractions to apply any reaction mechanism that is Instantaneous, Equilibrium, or Finite-Rate (for Mixture Fraction Approach). The Mixture Mass Fraction model 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 previous chart. If you would like to use one of these models, you must activate Species Mass Fractions.

Solve for Species Mass Fractions

Select Species Mass Fractions if you anticipate using one of the models available only for this approach during a later restart run. If you select Species Mass Fraction, you must select a Finite-Rate (for Species Fraction Approach) mechanism

The following chart shows the models that you can use with these options.

Mixture Mass Fraction Models Species Mass Fraction Models

Chemistry Chemistry

Finite-Rate (single step) Reactions Finite-Rate (multi step) Reactions

Instantaneous Reactions Surface Reactions

Equilibrium Reactions Properties (viscosity, conductivity) by Kinetic

Theory

Multi-Component Diffusion

Gas Phase Reaction*

Need text here.

Reaction Name*

Need text here.

Reaction Models (not for Eqlbm. Reaction)*

Equilibrium Products


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CO Oxidation Step

Thermal NOX

Nitrous NOX

Prompt NOX

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 N2 + O + M = N2O + M. 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

(70)

where the reaction rate kn is determined from kn = Ae-E/RT 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

(71)

where the reaction rate knp is determined from knp = Ae-E/RT 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.

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+.

(72)

The reaction rate kp is determined from kp = Ae-E/RT with A = 5.0 and E/R = 6000 in SI units. The O2 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. Fc 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 O2 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.

Effects of Turbulent/Chemistry Interaction 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

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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 unburned mixtures to 1 for burned 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

where SNO is the NO source

(73)

(74)

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

(75)

where 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+.

(76)

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 CO2, CO and OH have been calculated by CFD-ACE+ for the Warnatz CO oxidation option or that equilibrium values of CO2, CO, O2, and H2O 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 C3H8 (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

C0 + OH ↔ CO2+ H (77)


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From this reaction, the destruction of CO can be expressed as

(78)

where the reaction rate k1 is determined from , and A = 4.4e3, a - 1.5, and E/R = -373 in SI units. The subscript e indicates equilibrium concentrations. The constant A has been modified in CFD-POST to a value of 3.5e4 to better fit experimental results for practical combustors.

The oxidation of CO for the Dryer-Glassman option is given by

(79)

where the reaction rate k1 is determined from with A = 2.24e12, a - 10.0, and E/R = -20,000373 , and E/R=-20,000 in SI units. The constant A has been modified in CFD-POST to a value of 3.14e1 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.

Equilibrium Table Settings*

Min. Equi. Ratio

Max. Equi. Ratio

Table Settings*

Fuel Inlet Temp.

Oxid. Inlet Temp.

Pressure

Reference Pressure

CO Oxidation Step Coupling*

Coupled

Decoupled

Thermal NOX Model*

Simplified

Simplified Reversible

Extended

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MEDIA LIQUID PHASE

The Liquid Phase section of the Chem panel offers two applications: General Liquid Chemistry and Biochemistry.

GENERAL LIQUID CHEMISTRY*

Solve Concentration

Binary Diffusion*

Solvent Species

Molar Vol (Solvent)

Molar Vol (Solute)

Concentration Dependence*

Exponent (Alpha)

plus same as binary diffusion

Volume Reaction*

Volume Reaction Name

Define

General Liquid Chemistry Option

BIOCHEMISTRY

Binary Diffusion

Concentration Dependence

Solvent Species

Molar Vol (Solvent)


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Molar Vol (Solute)

Volume Reaction

Volume Reaction Name

Define

Ionization

Biochemistry Option

Liquid Phase Options Solve Concentration

To use this option, you must go to Tools menu > Database. When the Database Manager opens, click Species and define your species. Click Mixtures. Under User Input select Concentration. At the bottom of the window, select Enter Molar Concentration.

Binary Diffusion

Volume Reaction

Ionization

SPECIAL DISCUSSIONS

Two special situations are discussed here: Unsteady Combustion and In Situ Adaptive Tabulation (ISAT).

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. Follow these steps to set up the model.

1. On the PT panel select Chemistry and Turbulence modules. (Flow is automatically selected.)

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2. On the MO panel select the following:

1. Shared panel: Transient Conditions > Time Dependence > Transient.

2. Chemistry panel:

Chemistry Media > Media > Gas Phase

Gas Phase > Solve For > Species Mass Fractions

Gas Phase > Gas Phase Reaction.

Unsteady Combustion > Solve Combustion.


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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.

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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.

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.

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.

VOLUME CONDITIONS

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. Click the Volume Conditions [VC] tab to see the Volume Condition Panel.


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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. (As 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.

Note Density and viscosity properties are discussed in detail in Flow Module > Volume Conditions, and specific heat and conductivity properties are discussed in detail in Heat Transfer Module > Volume Conditions.

MASS DIFFUSION

The options available for Mass Diffusion vary depending on whether the Mixture Mass Fractions or Species Mass Fractions approach has been selected (see Solution Method). This section includes the following Mass Diffusion topics:

Constant Schmidt Number Constant Diffusivity Mix Polynomial in T Multi-Component Diffusion Chemistry VC Options

CONSTANT SCHMIDT NUMBER

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 (σ), the diffusion coefficient is calculated as:

(80)

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.

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.

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.

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(81)

The first part is the concentration-driven diffusion and is calculated as:

(82)

The second part is the thermo-diffusion or Soret diffusion and is calculated as:

(83)

The concentration-driven diffusion coefficient is then calculated as:

(84)

where

Dij =

σij =

σ = Lennard-Jones collision diameter ΩD = the collision integral

The collision integral, ΩD, is evaluated from the dimensionless temperature kB/εij where kB is Boltzmann's constant,

and εij is 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

(85)

where Kij is the thermo-diffusion ratio, and is given by


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where

Mi = molecular weight of species i T = temperature

σi

= characteristic diameter of the molecular

σij = 1/2 (σi+σj) Ω = collision integral

and Astr ,Bstr ,Cstr are some integral constant and given by

where Tst, dimensionless temperature constant, is given as Tst= k*T/e (k is Boltzmann's constant, T is temperature, and e is characteristic energy) and the index i depends on Tst, as shown here:

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All A0(i)…C3(i) are given constants:

A0(1) = 8.7140d-1, A1(1) = 8.5547d-1, A2(1) = 1.0310d+0, A3(1) = 4.0751d-1

B0(1) = 1.1719d+0, B1(1) = 7.5414d-1, B2(1) =-1.3929d+0, B3(1) = 6.6028d-1

C0(1) = 9.4008d-1, C1(1) =-4.5832d-1, C2(1) = 5.5896d-1, C3(1) =-2.0409d-1

A0(2) = 1.1253d+0, A1(2) =-3.1927d-2, A2(2) = 9.6617d-3, A3(2) =-8.7623d-4

B0(2) = 1.3878d+0, B1(2) =-2.8081d-1, B2(2) = 9.5936d-2, B3(2) =-1.1401d-2

C0(2) = 7.6201d-1, C1(2) = 9.0169d-2, C2(2) =-1.6830d-2, C3(2) = 1.0998d-3

A0(3) = 1.1253d+0, A1(3) =-3.1927d-2, A2(3) = 9.6617d-3, A3(3) =-8.7623d-4

B0(3) = 1.1255d+0, B1(3) =-1.2018d-2, B2(3) = 1.2583d-3, B3(3) =-3.6681d-5

C0(3) = 8.3975d-1, C1(3) = 3.4106d-2, C2(3) =-3.8932d-3, C3(3) = 1.5349d-4

A0(4) = 1.0929d+0,A1(4) = 2.1304d-3, A2(4) =-4.9384d-5, A3(4) = 4.3252d-7

B0(4) = 1.1255d+0, B1(4) =-1.2018d-2, B2(4) = 1.2583d-3, B3(4) =-3.6681d-5

C0(4) = 8.3975d-1, C1(4) = 3.4106d-2, C2(4) =-3.8932d-3, C3(4) = 1.5349d-4

A0(5) = 1.0929d+0, A1(5) = 2.1304d-3, A2(5) =-4.9384d-5, A3(5) = 4.3252d-7

B0(5) = 1.0950d+0, B1(5) = 0.0000d+0, B2(5) = 0.0000d+0, B3(5) = 0.0000d+0

C0(5) = 9.4335d-1, C1(5) = 1.6500d-4, C2(5) = 0.0000d+0, C3(5) = 0.0000d+0

The Multi-component Diffusion option also requires that you specify a method by which the program will satisfy species conservation, that is:

(86)

The three options available include the following:


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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 species (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.

CHEMISTRY VC OPTIONS

For a fluid volume, with VC Setting Mode set to Chemistry, select Chemistry Options.

VC Setting Mode - Chemistry – Chemistry Options

The Chemistry Options include the following:

Volume Reactions

This feature can be useful if you do not want the volumetric reactions in one of the fluid domains. Physically it only makes sense to disable volume reactions in a fluid region when it is disconnected from chemically reactive zones such as thin wall).

To activate this option, first select Gas Phase Reaction on the MO (model settings) panel and then select the volume. By default, Volume Reactions is enabled in all fluid domains. Switch between Enable and Disable as needed. Best practice is not having one fluid region with volume reactions Enabled joined by a common fluid/fluid interface to another fluid region whose volume reactions are Disabled.

Species Source You can specify a species source using this option. A species source can be specified using the user subroutine usource.

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Capillary Diffusivity

Although this option appears to be available, it is applicable only for Porous Media.

BOUNDARY CONDITIONS

Click the Boundary Conditions [BC] tab to open the BC panel. To assign boundary conditions and activate additional panel options, select an entity from the viewer window or BC Explorer.

The Chemistry module is fully supported by Cyclic, Thin Walls, and Arbitrary Interfaces boundary conditions.

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, and so forth).

This section covers the following topics: Inlets Outlets Walls Rotating Walls Symmetry Interfaces Thin Walls Cyclic Examples

INLETS

For any inlet boundary condition, you must specify how to set the species concentration for each cell face on the boundary condition patch.

Set species concentration for an inlet boundary condition

1. Under the Chemistry tab, select a mixture from the pull down menu (it lists all of the mixtures defined for the model).

2. If no mixture is present, click the Define button to launch the Property Manager and define a new mixture.

3. Under the Flow tab define the sub type of the inlet boundary condition.

4. A mass flow boundary condition will give the opportunity to define the mass flow rate depending on the flow rate settings in the database for the corresponding mixture (Option “from Mixture”).


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Note See Database Manager-Mixtures for details on how to define a mixture.

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 unmusical solution and convergence problems may result and you should relocate the outlet boundary condition to an area where there is total outflow if possible.

WALLS

Various boundary conditions can be specified under the Chemistry tab. They include the following:

Zero Flux Flux of all species to the walls is set to zero

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Fixed Mixture You can specify the mixture composition on all the cells adjacent to the wall. This mixture can be defined as “Constant” for both Gas and Liquid Chemistry, and, additionally, as “User Sub (ubound)” for Gas Chemistry. For “Constant” you need to define a mixture in the database manager which will show up in the list of mixtures available for this boundary. For “User Sub (ubound)” you also need to provide a mixture, which contains the species you are dealing with inside your user subroutine. The mass fractions/concentrations of the individual species in your mixture will be taken from the user subroutine. Therefore, you can define any arbitrary composition that sums to 1.

The definition of a fixed mixture at a wall will result into a diffusive species flux from the wall to the bulk flow. This flux will be printed into the Out file.

Species Specification

This option is valid only when Liquid Chemistry is selected and is similar to the Fixed Mixture boundary condition. You can pick each one of the species available and choose one of the four evaluation techniques in Evaluation method for that particular species. The species available can be defined in Tools > Active Mixtures & Species.

Surface Reaction

This option is available for both Liquid and Gas Chemistry. It allows for the definition of a surface reaction as defined in the database manager.

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 chemical conditions to that location in the same way as a you describe a Wall boundary condition.

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. A number of examples follow at the end of this section.

ROTATING WALLS

The Chemistry Module boundary condition specifications for rotating walls are identical to that as described for wall boundary conditions. They include the following:

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.

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.


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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.

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.

EXAMPLES

Dirichlet boundary condition at a fluid-solid interface

Dirichlet boundary condition at a wall using ubound user subroutine

Dirichlet boundary condition and the species flux output

Comparison of printed species flux in out file and result in CFD-VIEW

Example of Dirichlet boundary condition at a fluid-solid interface

Set-up in CFD-ACE-GUI

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Result in CFD-VIEW

Example of Dirichlet boundary condition at a wall using ubound user subroutine

When using a ubound user subroutine (please find coding example above) the user needs to define a mixture which includes all species dealt with in the user subroutine.


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Set-up in CFD-ACE-GUI

Result in CFD-VIEW. Result for bottom surface according to ubound.

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Example of Dirichlet boundary condition and the species flux output

FigSet-up in CFD-ACE-GUI

Species flux is well balanced:

Species Summary in out file


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Result in CFD-VIEW

Example of Comparison of printed species flux in out file and result in CFD-VIEW

For an inclined wall with a fixed species concentration the species flux to the bulk flow can be calculated with the calculator. For the 2d example below the calculator expression is:

chordlen()*average(sqrt(Diff_flux_x_H2O^2+Diff_flux_y_H2O^2))

The result of the species flux is here: 3.74E-05 kg H2O/s/m and is in good agreement with the printed species flux in the out file, which is 3.891235E-05 kg H2O/s/m.

For coarse meshes, you might get larger disagreements of the calculated fluxes in CFD-VIEW with the printed fluxes in the out file, especially if the neighboring boundary has a completely different value than the boundary of interest.

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Result in CFD-VIEW

Result in out file

INITIAL CONDITIONS

Click the Initial Conditions [IC] tab to open the Initial Condition panel.

You can specify initial conditions as constant values or read them from a previously run solution file. If you specify constant values, you must provide an initial mixture as required by the 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 the CFD-ACE+ User Manual > 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.

SOLVER CONTROL

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 Advanced Settings


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

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information on the different differencing schemes available. 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 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.


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.


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.

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.

When the Cut Diffusion option is selected for the Species Mass Fractions, the diffusive flux at the boundary faces is set to zero. Use the Acceleration option to control the other faces of the boundary cell. Values are between zero and one. By default this value is set to zero which means that the diffusive flux on all faces of the boundary cell is set to zero. When a value of one is used, the non-boundary fluxes are calculated normally. Acceleration values


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between zero and one linearly reduce the diffusive fluxes of the non-boundary faces. A value of zero helps convergence in some very low pressure cases.

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

1 More stability, Slower convergence

100 Less stability, Faster convergence


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.

OUTPUT

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 Summary Output

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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 -

Species Mole 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

SUMMARY OUTPUT

The species summary is written to the output file (modelname.out) and 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.

Under the Summary tab, select the summary 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.

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.

In addition to the summary species flow rate output, 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 Feature Scale Coupling under the Summaries 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 include the following:


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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.

POST PROCESSING

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. To do this, select CFD-VIEW (

) from the CFD-ACE+ toolbar to post-process the solutions.

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.

Post Processing Variables Variable Description Units

Nox_Rate NOx Production Rate kg/m3-s

Progress Progress Variable -

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React_Rate Reaction Rate (if gas phase reactions are present)

kg/m3-s

Species name Species Mass Fraction -

[species name]_mol_fraction Species Mole Fraction -

Dep_[species name] Deposition Rate (if surface reactions are present)

kg/m2-s

Diff_flux_x_[species name]

Diff_flux_y_[species name]

Diff_flux_z_[species name]

Total_flux_x_[species name]

Total_flux_y_[species name]

Total_flux_z_[species name]

Species Flux kg/m2-s

D_[species name] Species Diffusivity m2/s

DT_[species name] Species Thermodiffusivity m2/s


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?


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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)

References

Kerstein, A. R. 1988. "A linear eddy model of turbulent scalar transport and mixing." Combust. Sci. Tech. 60:391.

McMurtry, P.A., S. Menon, and A. R. Kerstein. 1992."A linear eddy sub-grid model for turbulent reacting flows: application to hydrogen-air combustion." Twenty-Four Symposium (International) on Combustion, The Combustion Institute :271-278.

Pope, S. B. 1997. "Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation.” Combustion Theory and Modeling, 1: 41-63.

Pratt, D. T. and J.J. Wormeck. 1976. "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.

WORKS CONSULTED

Hirschfelder, J O., C. F. Curtiss, and R. B. Bird. 1954. Molecular Theory of Gases and Liquids. John Wiley & Sons, Inc., New York.

Somorjai, G.A. 1994. Introduction to Surface Chemistry and Catalysis. Wiley-Interscience, New York.

User Scalar Module

Overview

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:

Applications Features

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Theory Limitations Implementation Frequently Asked Questions References

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).

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.

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.

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 density is taken out of


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the generic transport equation. One typical example of this type of scalar is the electric potential.

Limitations The User Scalar module is currently not compatible with the Free Surfaces Module (VOF).

Theory

The generic transport equation of a user scalar φ is written as:

(5-1)

where D is the diffusivity and Sφ 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).

At a solid/solid or fluid/solid interface, the diffusive flux normal to the boundary is conserved:

(5-3)

Implementation Overview

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.

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).

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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.

MODEL SETUP AND SOLUTION OVERVIEW

CFD-ACE+ provides the inputs required for the User Scalar Module. Model Setup and Solution requires data for the following panels:


PROBLEM TYPE

Click the Problem Type [PT] tab to see the Problem Type Panel. See Control Panel-Problem Type for details. Select User Scalar to activate the User Scalar Module. The User Scalar Module can work with any of the other modules in CFD-ACE+.

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.

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).

4. To create a General Scalar, activate both Convection and Diffusion in Solid.


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5. To create a Passive Scalar, activate only Convection.

6. To create a Poisson Scalar, activate only Diffusion in Solid.

7. Enter a name for the current scalar in the Scalar Name field.

8. Proceed with step 2 for every scalar.

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.

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:

3. Constant-D = value specified in m2/s.

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4. Schmidt Number - D = μ/Sc

5. User Sub (udiff_scalar) - D is defined by a user subroutine (udiff_scalar). Please see User Subroutines for details.

6. Enter the value of diffusivity or Schmidt number as appropriate.


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 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, and 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

Desired Effect Coefficient Setting

Result a b c

Fixed Value (Dirichlet) a=0 b≠0 c=c

Fixed flux (Neumann) a≠0 b=0 c=c


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Flux as Function of Value (Combined)

a≠0 b≠0 c=c

A typical input panel for any user scalar boundary condition is shown.

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.


INITIAL CONDITIONS


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.

SOLVER CONTROL

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.

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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.



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The Solver Control section includes the following:

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.

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.

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.

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.

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.

Summary Output Under the Summary tab on the Solution Control page, select the summary 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 summary output options. See Control Panel-Summary Output for details on the general summary output options including boundary 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

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Variable Units

Density kg/m3

Scalar Values scalar units

Scalar Diffusivity kg/m-s (for passive or general scalar)

m2/s (for poisson scalar)

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.

Post Processing Variables Variable Description Units

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.


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.


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References

Versteeg, H.K. and Malasekera, W., 1995, "An Introduction to Computational Fluid Dynamics." John Wiley & Sons Inc, New York, pp24.

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 mm 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 topic includes the following sections:

Check that these links are working:

Applications Theory

Blackbody Radiation Radiation Properties Radiation Characteristics of Gases Radiative Transfer Equation (RTE)

Solution Methods Surface-to-Surface Discrete Ordinate Method Monte-Carlo Method P1 Method

Radiation Properties Absorption Coefficient Sets Emissivity Sets Scattering Coefficient Sets Spectral Refractive Index Sets Specularity Sets

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Post Processing Frequently Asked Questions References

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 do the following:

• 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. • Solve radiative heat transfer problems using one of the available 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.

Theory

check that these links work.

INTRODUCTION TO RADIATION THEORY

This section includes the following topics:

Blackbody Radiation Radiation Properties Radiation Characteristics of Gases Radiative Transfer Equation (RTE)

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, for example, Siegel and Howell, 1992:

Eq 1


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where

Eq 2

where h is Planck’s constant (6.6260755x10-34 Jsec), k is Boltzmann’s constant (1.380658x10-23 J/K), λ is the wavelength of radiation, c is the speed of light, n is the refractive index of the medium into which emission occurs, all with reference to vacuum, and T is the absolute temperature. The previous equation is independent of the nature of the material emitting radiation. The following equation 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

Eq 3

which is called the Wien’s displacement law. Integrating equation 6-1 over all the wavelengths, assuming n is spectrally constant, results in the Stefan-Boltzmann law given by

Eq 4

where σ is the Stefan-Boltzmann constant (e5.67051x10-8 W/m2K4).

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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 [λ1, λ2] can be obtained analytically using a series approximation developed by Chang and Rhee:

Eq 5


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where:

Eq 6

RADIATION PROPERTIES

All real substances do not absorb or emit as blackbodies. The emissive power of an arbitrary surface at temperature T into a medium of refractive index n is given by

Eq 7)

where ε is the total hemispherical emissivity of the surface and 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 spectral hemispherical emissive power of a surface is given by:

Eq 8

where ελ is called the spectral hemispherical emissivity. The total emissivity and spectral emissivity are related by the following equation:

Eq 9

When radiation is incident on a surface, some of the energy is absorbed, some reflected and the rest transmitted. This behavior is characterized by: absorptivity (α), defined as the fraction of incident energy that is absorbed; reflectivity (ρ), defined as the fraction of energy reflected; and transmissibility (τ) defined as the fraction of energy transmitted. Clearly, the sum of these quantities is unity, that is:

Eq 10

For an opaque surface, the transmissivity is 0 and hence

Eq 11

Kirchhoff’s law states that at thermal equilibrium, the emissivity of a surface is equal to the absorptivity, that is:

Eq 12

Combining the previous two equations:

Eq 13

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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 (εg). 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:

Eq 14

where Lm is the mean beam length which may be calculated (for optically thin gas radiating to its entire boundary) as:

Eq 15

where V is the volume of the enclosure and A is the area of the boundaries.

RADIATIVE TRANSFER EQUATION (RTE)

The integro-differential radiative heat transfer equation for an emitting-absorbing and scattering gray medium can be written as

Eq 16

where Ω is the direction of propagation of the radiation beam, I is the radiation intensity which is a function of both position (r) and direction (Ω), κ and σ are the absorption and scattering coefficients respectively, Ib is the intensity of black body radiation at the temperature of the medium and Φ is the phase function of the energy transfer from the incoming Ω' direction to the outgoing direction Ω. The term on the left-hand side represents the gradient of the intensity in the specified direction Ω. 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 here.


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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:

Eq 17

where I is the intensity of radiant energy leaving a surface at a boundary location, ε is the surface emissivity, ρ is the surface reflectivity, and n is the unit normal vector at the boundary location.

Solution Methods

INTRODUCTION TO SOLUTION METHODS

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 and are described in this section:

Surface-to-Surface Discrete Ordinate Method Monte-Carlo Method P1 Method

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Activate the Radiation Module from the PT panel. Use the MO panel > Rad tab to select the solution method of choice.

SURFACE-TO-SURFACE (STS) METHOD

This section covers the following topics for the STS method.

Theory Limitations Grid Requirements Implementation Option Panels

STS THEORY

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).

Eq 18

The integral formulation of the above equation is

Eq 19

where:

Es = σT4 = the blackbody emission power

σ = the Stefan-Boltzmann constant

qs = the surface radiation flux

g = Es-(1-ε)/ε = the radiosity

n and n' are normal at r and r', respectively

The kernel, K , is defined as:

where υ is the visibility function defined as:


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Equation 6-19 can be written as

Eq 20

where is the location hit by the ray emitted from r in the Ω direction.

The angular integral on the right hand side of equation 6-20 can be replaced by numerical quadrature of the form

Eq 21

where θ is the angle between the normal and Ω and M are the number of angular integration points. Then the discrete form of equation 6-20 is:

Eq 22

where r'ij is the hit point of the ray emitted from r'i in Ωj.

STS LIMITATIONS

The Surface-to-Surface model does not account for any participating medium; hence, radiation through semi-transparent solids cannot be handled. This model supports gray radiation only and does not work with cyclic boundary conditions.

STS GRID REQUIREMENTS

This method can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). In addition, all grid cell types are supported (quad, tri, hex, tet, prism, poly). Note that this method does not require any volume mesh between the boundaries, and will not interact with it if it exists.

STS IMPLEMENTATION

After selecting Radiation (Rad) on the PT panel of the Control Panel, select your model’s options and then move through the remaining tabs to completely define your model. This section describes the various Control Panel options available.

Note Gray and Non-Gray options are not supported by the STS model.

STS OPTION PANELS

MO PANEL

The MO (model options) panel contains three tabs on its left side. The Shared tab contains no settings that affect the Radiation model, and the Advanced (Adv) tab contains no options for STS. Use the Radiation (Rad) tab to set most of the specifications for your model.

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The following image shows the MO panel with the Rad tab selected. In this image the Surface to Surface model and Solar Irradiation option are selected. If Solar Irradiation is not selected, then Specify Radiation Sources is unavailable.

Model Options Panel (Rad tab) with STS and Solar Irradiation Selected

See Also Absorption Coefficient Sets Emissivity Sets

Rad Tab Options (STS MO Panel) Model Select Surface to Surface from the available options.

Accuracy You can select a level of accuracy from the following choices: 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.

Subiteration Determines 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.

Environment Temperature

Use if any of the surfaces can see outside of the computational domain. This may happen if the domain is not closed, if some surfaces are transparent, or if some surfaces 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.


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Solar Irradiation Enables you to model point radiation sources. Once this feature has been activated, you will be able to Specify Radiation Sources that are present. A radiation source can be used to model solar radiation.

Radiation Sources Dialog Box

Use the Radiation Sources panel as follows:

1. Set the number of Total Sources and click OK.

2. For each radiation source, give a direction and an intensity value. The direction is specified by supplying a normal vector which gives the direction of the incoming radiation. The radiation source itself is assumed to be located an infinite distance away from the global coordinates system 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).

3. Click Insert to add a new row above the active row.

4. Click Append to add a new row to the end of the table.

5. Click Delete to remove the highlighted row.

VC PANEL (STS)

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. Depending on the property values selected, various options are available. For more information on the VC panel options, please refer to ACE User Manual > CFD-ACE-GUI > Control Panel > Volume Conditions.

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VC Panel > Rad Tab (STS)

Media Participation Select one of the Absorption Coefficient sets (which have been defined under MO panel > Rad tab).

For the STS method, the value of the set acts as a flag to let the solver know whether the volume is transparent or opaque. Acceptable set values are 0 (transparent) and -1 (opaque).

Default Emissivity Set Select one of the Emissivity sets (which have been defined under MO panel > Rad tab).

Even though emissivity is a boundary property, you are given the opportunity to select an emissivity set at the VC level for ease of use. Indeed, the emissivity set chosen here will be applied to all of this volume’s boundaries for which the emissivity option is “From VC” (to be selected under BC panel > Rad tab of each individual boundary). This allows fast and easy setup of boundary emissivity if the same value is to be applied at many boundaries.

If the emissivity option of an interface boundary is set to “From VC”, and the two volumes that share this boundary have a different emissivity set selected, the emissivity of the opaque volume will be applied to the interface boundary.

See Also Emissivity Sets

BC PANEL (STS)

Each boundary condition is assigned a type (for example, Inlet, Outlet, Wall, and so forth). Because radiation is a ray-based phenomenon, a radiation boundary condition must be given for all of the computational boundaries in the model including inlets and outlets.


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All of the general boundary conditions for the Radiation module are located under the BC radiation (Rad) tab and can be reached when the BC Setting Mode is set to General and a Boundary is selected from the Viewer Window or the BC Explorer. For more information on the BC panel options, please refer to ACE User Manual > CFD-ACE-GUI > Control Panel > Boundary Conditions.

BC Panel > Rad Tab (STS Method)

Surface Normal If Surface Normal is set to Inward, then the computational boundary is exchanging radiative information with the interior (volume) of the computational domain.

If Surface Normal is set to Outward, then the computational boundary is exchanging radiative information with the external environment: it can exchange radiation fluxes with the environment based on the Environment Temperature (as defined under MO panel > Rad tab > Environment Temperature), as well as with other Outward boundaries from domains not attached to the current one (that is, no grid exists between either boundaries of each volume).

Surface Type Use Surface Type to specify an Opaque or a Transparent surface type.

Note The Surface Normal and Surface Type options appear for all boundary types except the Interface type.

Emissivity Set Select From VC or one of the emissivity sets (which have been defined under MO panel > Rad tab. If From VC is selected, the emissivity value will be taken from the set chosen under VC panel > Rad tab > Default Emissivity Set.

See Also Emissivity Sets

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IC PANEL (STS)

The STS model requires no settings here.

SC PANEL (STS)

The STS model requires no settings here.

OUT PANEL (STS)

The STS model does not have any specific output.

Note on .sts.rst File - When the Restart option is checked on the Out panel > Restart tab, a modelname_sts.rst file is created at the end of the solution. This file consists of all the intermittent data required to calculate form factors. The time required to calculate this data is huge. When the user wants to restart the DTF, the solver directly reads this file from the working directory, and the time to recalculate the intermittent data is thus reduced and the process becomes faster. In the absence of an *_sts.rst file at the time of restart, the solver writes a warning message in the OUT file informing that the *_sts.rst is not available, and the simulation will continue to run by generating the file again.

RUN PANEL (STS)

The Run panel allows you to control start/stop CFD-ACE-Solver and monitor the simulation while it is running. For a complete description, please see ACE User Manual > CFD-ACE-GUI > Control Panel > Run Controls.

DISCRETE ORDINATE METHOD INTRODUCTION

Two different approaches are available in CFD-ACE+ for the solution of Radiative Transfer Equation (RTE) in general participating media. These are the Sn Discrete Ordinate Method (SnDOM) (Fiveland, 1988) and the Control Angle Finite Volume Method (CAFVM) (Raithby and Chui, 1990; Chai, Lee and Patankar, 1994). They are collectively referred to as discrete ordinate methods (DOM) in CFD-ACE+. A detailed description of the implementation of these methods for multi-dimensional radiation in unstructured grids in the context of CFD-ACE+ is available (Vaidya, 1998).

DOM THEORY

This section describes the following topics:

Wall Boundary Symmetry Boundary Inlet Exit Boundary Fresnel Interface Boundary Conjugate Heat Transfer AAQ Model

The Sn quadrature schemes available in CFD-ACE+ are: S4, S6, S8, and S12. The number of ordinate directions used in the radiation computations when using the above schemes are, respectively, 24, 48, 80, and 168 in three-


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dimensional problems, and one half of those in the case of two-dimensional problems with or without axisymmetry. In CFD-ACE+, the S4 approximation, which considers 12 ordinate directions in two dimensions (24 in three dimensions) is currently the default option. The selection of ordinate directions is not arbitrary but must satisfy the symmetry and moment invariance constraints.

In CAFVM, since the computational directions can be user specified, it requires the specification of the number of cells N and N in the polar and azimuthal directions, respectively, in the discretization of a unit sphere as shown here.

Unit Sphere discretization in CAFVM

Note that the user specified Nθ and Nφ are taken to refer to the total number of control cells in the discretization of a full unit sphere. The spherical coordinate extents and the total number of directions used in the computations are:

• 3D: (0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π)

Total number of computed directions = Nθ Nφ

• 2D: (0 ≤ θ ≤ π/2, 0 ≤ φ ≤ 2π)

Total number of computed directions = (Nθ ⁄ 2) Nφ

• 2D Axisymmetric: (0 ≤ θ ≤ π, 0 ≤ φ ≤ π)

Total number of computed directions = Nθ (Nφ /2)

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:

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Eq 23

In the previous equations, m and m' denote the outgoing and incoming directions, respectively. For a direction m, Wm represents the associated weight while α, β, and γ 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.

For the gray model, the subscript λ should be dropped from the above equation and ΔF becomes unity. In the non-gray model, the radiative properties are assumed to be functions of wavelength 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.

The in-scattering term on the right-hand side of equation 6-23 contains the phase function φ 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

Eq 24

where a0is 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’. (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:

Eq 25

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 kλ= 0 .

WALL BOUNDARY

The outgoing intensity at an opaque wall is the sum of the intensity contributions due to emission and reflection from the wall:

Eq 26


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At an adiabatic wall, the wall temperature is calculated by balancing the radiative and conductive heat fluxes. The specular ρs and diffuse ρd components of reflectivity can be specified using the specularity parameter β as follows:

Eq 26a

SYMMETRY BOUNDARY (SPECULAR REFLECTION)

A symmetry boundary is a purely specularly reflecting surface:

Eq 27

INLET EXIT BOUNDARIES

At these boundaries, as in the case of an opaque wall, we have:

Eq 28

When the boundary faces are not aligned with the specified angular discretization of a unit sphere, errors arise due to truncated solid angles. In the CAFVM approach, where each discrete direction can be associated with a finite solid angle, that is, a surface patch on a unit sphere, such errors can be minimized by dividing the patch into finer subdivisions (see for example, Vaidya 1998), as illustrated in the following figure. Currently, this numerical approach for achieving increased accuracy is available for the CAFVM approach only and not for SnDOM.

Unit sphere subdivision at a Boundary Face

FRESNEL INTERFACE BOUNDARY

At a Fresnel interface, which is the interface between media of different refractive indices, radiation incident on the interface is both reflected and transmitted.

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Optics at a Fresnel interface

When the propagation is from a medium of higher refractive index, the radiation from that side of the interface undergoes Total Internal Reflection (TIR) and zero transmission when the incident angle is greater than the critical angle. In general, the reflected and transmitted directions are not expected to coincide with the specified discretization directions. A subdivision approach (Murthy and Mathur, 2000) is used for computing these reflected and transmitted fluxes, with the implementation in CFD-ACE+ (Vaidya, 2011) designed for improved accuracy. Currently, Fresnel interface boundaries can be handled only in the CAFVM and not SnDOM approach.


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Unit Sphere Subdivision Configuration

The transmitted intensity direction at a Fresnel interface is obtained using Snell’s law while the reflected direction is obtained using the law of specular reflection. The spectral reflectivity and transmissivity optical properties can be computed from a plane wave electromagnetic analysis. In the current implementation, assuming the Fresnel interface to be an ideal dielectric-dielectric interface and the radiation to be unpolarized, the spectral reflectivity for incidence from medium 1 to medium 2 is computed as

Eq 29

where and are the reflection coefficients for parallel and perpendicular polarizations, respectively, expressed as:

Eq 30

Here, n1 and n2 are the refractive indices of the media, θ1 is the incident, and θ2 is the transmission angle.

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, that is

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Eq 31

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

Eq 32

where

Eq 33

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 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:

Eq 34

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 .0and 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.

AAQ MODEL

The AAQ model provides a more conservative formulation in calculating the incidence radiative flux on a wall for general 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 π.

DOM LIMITATIONS

The Discrete Ordinate Method, in general, is comparatively not very accurate for optically thin media. The SnDOM Discrete Ordinate Method, in contrast to the CAFVM method, has problems with specular radiation. The ordinate set implemented in the SnDOM method 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.


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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.

DOM GRID REQUIREMENTS

This method can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). In addition, all grid cell types are supported (quad, tri, hex, tet, prism, poly).

The general grid generation concerns apply; that is, ensuring that the grid density is sufficient to resolve solution gradients, minimizing skewness in the grid system, and locating computational boundaries in areas where boundary values are well known.

DOM IMPLEMENTATION

After selecting Radiation (Rad) on the PT tab of the Control Panel, move through the remaining tabs to select and define your model. This section describes the various options available.

DOM OPTION PANELS

MO (MODEL OPTIONS) PANEL

The MO (model options) panel contains three tabs on the left side of the panel. The Shared tab contains no settings that affect the Radiation model. Use the Radiation (Rad) tab to set most of the specifications for your model. Use the Adv tab to set additional properties for the selected DOM Option (SnDOM or CAFVM).

These options refer to the Sn Discrete Ordinate Method (SnDOM) and the Control Angle Finite Volume Method (CAFVM), collectively termed simply as discrete ordinate methods. You can specify unique emissivity sets and absorption coefficient sets with both SnDOM and CAFVM options. If you choose the CAFVM option, you can specify refractive index and specularity sets. Usually each boundary surface in the simulation requires specification of emissivity and specularity sets, and each solid or fluid volume requires absorption coefficient and refractive index sets.

You can use the DOM method in the gray (wavelength independent properties) or non-gray (wavelength dependent properties) mode. To use the gray mode, simply ensure that the Non-Gray box is unchecked. If you activate Non-Gray, you must input the Number of Bands and Specify Wavelength Bands to be used for radiative property specifications. DOM Options vary based on the Model and DOM Options selected.

The following image shows the MO panel of the Control Panel with the Rad tab selected. The following images show the various combinations available for Model, DOM options, and gray properties.

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Discrete Ordinate Method (DOM), SnDOM, Non-Gray

Discrete Ordinate Method (DOM), SnDOM, Gray


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Discrete Ordinate Method (DOM), CAFVM, Non-Gray

Discrete Ordinate Method (DOM), CAFVM, Gray

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See Also Absorption Coefficient Sets Emissivity Sets Scattering Coefficient Sets Spectral Refractive Index Sets Specularity Sets

Rad Tab Options (DOM MO Panel) Model Select Discrete Ordinate Model from the available options.

DOM Options Select SnDOM or CAVFM.

Non-Gray Check this box for Non-Gray options:

Number of Bands - Specify the number of bands.

Specify wavelength bands – Enter the wavelengths in μm.

Emissive Power (User sub(uradiation))

Adv Tab Options (DOM MO Panel) SnDOM

In the Quadrature Scheme pull-down menu, specific quadrature schemes of the Sn discrete ordinate method can be selected. These quadrature schemes are: S4 (default), S6, S8, and S12. The number of ordinate directions used in the radiation computations when using the above schemes are, respectively, 24, 48, 96, and 168 in three-dimensional problems, and one-half of those in the case of two-dimensional problems with or without axisymmetry. Higher-order schemes such S8 and S12 are particularly more demanding on computational resources because of the increased number of directions that need to be stored and solved.


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CAVFM

Because the computational directions can be user-specified, the specification of the number of cells N and N in the polar and azimuthal directions, respectively, are required in the discretization of a unit sphere. Note that the user-specified N and N are taken to refer to the total number of control cells in the discretization of a full unit sphere. The spherical coordinate extents and the total number of directions used in the computations include the following:

• 3D: (0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π) Total number of computed directions = Nθ Nφ

• 2D: (0 ≤ θ ≤ π/2, 0 ≤ φ ≤ 2π) Total number of computed directions = (Nθ ⁄ 2) Nφ

• 2D Axisymmetric: (0 ≤ θ ≤ π, 0 ≤ φ ≤ π) Total number of computed directions = Nθ (Nφ /2)

VC PANEL (DOM)

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. Depending on the property values selected, various options are available. For more on the VC panel options, please refer to ACE User Manual > CFD-ACE-GUI > Control Panel > Volume Conditions.


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BC PANEL (DOM)

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 describes the implementation of each BC 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.

See Also Radiation Properties

SC PANEL (DOM)

The Solver Control (SC) panel provides access to the various settings that control the numerical aspects of the CFD-ACE-Solver and all of the output options.

Iter Use the Iter tab to specify the number of times the DOM 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.

Other tabs Refer to user manual

For more on the SC panel options, please refer to ACE User Manual > CFD-ACE-GUI > Control Panel > Solver Controls.

OUT PANEL (DOM) Graphical Output Use the Graphics tab to 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.

Variable Units

Radiative Wall Heat Flux W/m2

Summary Output Use the Summary tab to select the summary 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 summary output options. (See Summary Output for details on the general summary output options including boundary 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


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conduction/convection.

RUN PANEL (DOM)

The Run panel enables you to control the CFD-ACE-Solver and monitor the simulation while it is running. For a complete description, please see ACE User Manual > CFD-ACE-GUI > Control Panel > Run Controls.

MONTE CARLO METHOD INTRODUCTION

The Monte Carlo (MC) 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.

The Monte Carlo model computes radiative properties from first principles. In addition, it accepts surfaces of many different reflection characteristics (for example, diffuse, specular, and 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.

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.

MC THEORY

This section describes the following topics:

Radiative Transfer Equation Monte Carlo Raytracing Patch Definitions Radiative Properties

RADIATIVE TRANSFER EQUATION

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 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

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Eq 35

where:

Qi = Heat Flux (W)

qi = Heat Flux density (W/m2)

Ai = Area (m2)

δij = Kronecker delta

εj = Emissivity of Patch j

Rij = Radiation exchange matrix (fraction of radiation emitted by patch i and absorbed by patch j

σij = Stefan-Boltzmann constant (5.669 x 10-8 W/m2-K4)

Ti = Average temperature of patch j (K)

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).

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 five MC updates during the whole run. 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.

PATCH DEFINITIONS

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. In the current setup, we employ a simple ASCII file to set up and input these radiative 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?.

In principle, it is possible to define each boundary cell face as a patch. This, however, leads to two problems:

1. 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.


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2. Some boundary cell faces may be extremely small. This will result 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.

Using the previous patch definition, the following conclusion may be drawn, and should be kept in mind while defining patches:

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 heat flux.

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.

RADIATIVE PROPERTIES

As 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 Azzam and Bashara, 1977).

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? For optical properties of materials (Palik, 1985).

When you choose the Monte Carlo method, the Rad 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.

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The Monte Carlo model computes radiative properties from first principles. In addition, it accepts surfaces of many different reflection characteristics (for example, diffuse, specular, and 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.

In the current setup, we employ a simple ASCII file to setup and input these radiative 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?.

MC LIMITATIONS

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.

MC GRID REQUIREMENTS



MC IMPLEMENTATION

After selecting Radiation (Rad) on the PT tab of the Control Panel, move through the remaining tabs to select and define your model. This section describes the various options available.

MC OPTION PANELS

MO PANEL (MC)

The MO panel includes the Share, Rad, and Adv tabs on the left side of the panel. The Shared tab contains no settings that affect the Radiation model. Use the Radiation (Rad) tab to set most of the specifications for your model. You do not use the Adv tab with the Monte Carlo model.

When you choose the Monte Carlo method, the Rad tab is as shown in the following figure. 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.


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MO > Rad Tab > Monte Carlo Model

Rad Tab Options (MC MO Panel) Number of Rays 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. The Monte Carlo model computes radiative properties from first principles. In addition, it allows surfaces of many different reflection characteristics (for example, diffuse, specular, and 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.

Solution Skipping Frequency

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 five MC updates during the whole run. This means that the Solution Skipping Frequency will be 100. To enter this value, select Iteration and enter the Value.

Statistics Model In CFD-ACE+ Monte Carlo calculations are performed using two kinds of statistical formulations. The Pseudo MC approach employs pseudo random numbers from a uniform deviate, while the Quasi Monte Carlo 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.

VC PANEL (MC)

You do not need to specify any volume conditions for the Monte Carlo model.

BC PANEL (MC)


The Radiation module handles boundary conditions slightly differently than most of the other modules, because radiation is a ray-based phenomena. Therefore, a radiation boundary condition must be given for all of the

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computational boundaries in the model. All of the general boundary conditions for the Radiation module are located under the Radiation (Rad) tab on the BC panel and can be reached when the BC setting mode is set to General. Each boundary condition is assigned a type (for example, Inlet, Outlet, Wall, and so forth).

BC Panel: Rad Tab Options BC Setting Mode

BC Type

Rad type

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.

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 temp - 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


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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 its 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.

No of sub-patches – need text here

IC PANEL (MC)

The Monte Carlo model requires no settings here.

SC PANEL (MC)

The Solver Control panel allows access to the various settings that control the numerical aspects of the CFD-ACE-Solver and all of the output options. In general, there are no special numerical control settings required in the Radiation module, but users have an option for using enhanced specular computation leading to better radiation energy balance.

Enhanced Accuracy Specular Computation

This option enables better computation of specular direction during Total Internal Reflection, when a ray undergoes a second refraction. This calculation leads to a lesser number of lost rays, enhancing better radiation energy balance.

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Option to set enhanced accuracy for specular reflection

OUT PANEL (MC)

There are no settings under the Output tab for other radiation models.

RUN PANEL (MC)

The Run panel enables you to control the CFD-ACE-Solver and monitor the simulation while it is running.

P1 METHOD INTRODUCTION

Instead of directional discretization, the P1 model assumes 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


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substituted into the radiative transfer equation (RTE) and manipulated, results in a set of coupled diffusion-like equations.

P1 THEORY

This section also covers the following topics:

Coupling to Overall Energy Equation Treatment of Non-Gray Radiation: The Stepwise Gray Model

When only the leading term in the series is retained (that is, N = 1 in the PN expansion), the result is a single Helmholtz equation, the P1 equation, which is written as

(1)

subject to the boundary condition

(2)

where ελ is the emissivity of the boundary surface having surface normal . The quantity βλ ( = κλ+σλ ) is also known as the spectral extinction coefficient.

The quantity Gλ is known as the spectral integrated intensity or incident radiation, and is defined as .

Ebλ( = π Ibλ ) 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. After G λ has been obtained by solving equation 1, the divergence in the radiative heat flux can be computed using the following relation

(3)

where q is the radiative heat flux vector. Also, of practical interest is also the radiative heat flux normal to any surface. This may be written as:

(4)

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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 Ebλ is replaced by the following:

.

In the non-gray case, special treatment of equation 1 is necessary. Integration of equation 1 over the entire spectrum yields the following:

(5)

In the step-wise gray model, the spectrum is first split into discrete spectral intervals termed bands. Within each band, k, the radiative properties (that is, βλ, κλ, σs λ ) are assumed to be constant. Under this approximation, equation 5 may be re-written as a set of gray equations

(6)

where βk and κk are the extinction and absorption coefficients of the kth band, respectively. The total number of bands is denoted by Nb. The quantities Gk and Ebk represent the incident radiation and blackbody emissive powers within the kth 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.

P1 LIMITATIONS

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.


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P1 GRID REQUIREMENTS



P1 IMPLEMENTATION

The MO panel includes the Share, Rad, and Adv tabs on the left side of the panel. The Shared tab contains no settings that affect the Radiation model. Use the Radiation (Rad) tab to set most of the specifications for your model. The Adv tab is not used for P1 modeling.

P1 OPTIONS PANELS

MO PANEL

The following image shows the MO panel of the Control Panel with the P1 model and Rad tab selected.

Figure 14 Model Options Rad Panel with P1 Selected

Rad Tab Options (P1 MO Panel) MO Panel (P1): Rad Tab Options

Non-Gray - Requires that the number of bands be specified.

Number of Bands - Specify the number of bands.

Emissive Power - (User Sub(uradiation)


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VC PANEL (P1)

You do not need to specify any volume conditions for model types.

BC PANEL (P1)


The Radiation module handles boundary conditions slightly differently than most of the other modules, because radiation is a ray-based phenomena. Therefore, a radiation boundary condition must be given for all of the computational boundaries in the model. All of the general boundary conditions for the Radiation module are located under the Radiation (Rad) tab on the BC panel and can be reached when the BC setting mode is set to General. Each boundary condition is assigned a type (for example, Inlet, Outlet, Wall, and so forth).

IC PANEL (P1)

This model requires no IC settings.

SC PANEL (P1)

The Solver Control (SC) panel allows access to the various settings that control the numerical aspects of the CFD-ACE-Solver and all of the output options. The SC panel allows access to the various settings that control the numerical aspects of the CFD-ACE-Solver and all of the output options.

OUT PANEL (P1)

This model requires no Out settings.

RUN PANEL (P1)

The Run panel enables you to control the CFD-ACE-Solver and monitor the simulation while it is running.

Radiative Properties

RADIATIVE PROPERTIES OVERVIEW

Depending on the model option you select, some radiative property data is required. The following chart identifies the available properties for the various models. The Monte Carlo model does not use these options.

Radiative Properties

Options

DOM

SnDOM

DOM

CAVFM P1 STS

Gray Non-

Gray Gray

Non-

Gray Gary

Non-

Gray Gray

Absorption Coefficient Sets x x x x x x x


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Emissivity Sets x x x x x x x

Scattering Coefficient Sets x x

Spectral Refractive Index

Sets

x x

Specularity Sets x x

ABSORPTION COEFFICIENT SETS

Absorption coefficient set data is required for DOM, P1, and STS. For Gray property settings the absorption coefficient values will be the same for all wavelengths (that is, Band 1 = 0 - ∞ μm). For DOM and P1 Non-Gray property settings, set the number of bands and their wavelengths on the MO > Rad tab.

For the DOM and P1 methods, the absorption coefficient set determines if a volume condition is opaque, transparent, or semi-transparent. For the STS method, the absorption coefficient set is only used as a switch to determine if a volume condition is opaque or transparent. The following chart summarizes these values.

Absorption Coefficient Set Values

Opaque Transparent Semi-transparent

DOM or P1 -1 0 greater than 0

STS -1 0 N/A

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. Determining which set to apply to each volume condition will be easier if you use material names for the sets.

The following image shows the Specify Absorption Coefficient Sets dialog box . Insert adds a new row above the active row. Append adds a new row to the end of the table. Delete removes the highlighted row. Although the first three rows of the table may not be manually altered, you can give additional sets a meaningful name simply by typing over the default name of Set n.

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Absorption Coefficient Sets Dialog Box

EMISSIVITY SETS

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 determines the emissivity of the opaque surface.

The value is the total hemispherical emissivity and must range between 0 and 1. When working with gray property settings, the emissivity value will be the same for all wavelengths (that is, Band 1 = 0 - ∞ μm). For DOM and P1 Non-Gray property settings, set the number of bands and their wavelengths on the MO > Rad tab.

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. Determining which set to apply to each boundary condition will be easier if you use material names for the sets.

Insert adds a new row above the active row. Append adds a new row to the end of the table. Delete removes the highlighted row. The first two sets of the table may not be manually altered.

Emissivity Sets Dialog Box


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SCATTERING COEFFICIENT SETS

Scattering coefficient data is required only for DOM with the CAFVM option (Gray and Non-Gray).

There is one default set name, Non-Scattering, which specifies the scattering coefficient value zero. For other scattering coefficient sets, you may give a name and a value. The name will be used later when assigning radiative properties to volume conditions. Determining which set to apply to each volume condition will be easier if you use material names for the sets.

The scattering coefficient value for Gray properties will be the same for all wavelengths; that is, Band 1 = 0 - ∞μm. However, if you are using the Non-Gray option, the scattering coefficient value may be different for each wavelength band. You set the total number of bands on the MO > Rad tab, as well as the range of band wavelengths.

You specify the total number of scattering coefficient sets in the Number of sets box on the Scattering Coefficient Sets dialog box. Click Update and the appropriate number of new rows is added.

Insert adds a new row above the active row. Append adds a new row to the end of the table. Delete removes the highlighted row. The first two rows of the table may not be manually altered.

Scattering Coefficient Sets Dialog Box

SPECTRAL REFRACTIVE INDEX SETS

Spectral refractive index data is required only for the DOM method with CAFVM option (Gray and Non-Gray). At the interface separating media of different refractive indices, Snell and Fresnel optical relations are used to compute reflected and transmitted radiation directions and fluxes.

Specify the total number of spectral refractive index sets in the Number of sets box. For each spectral refractive index set, you may give a name and value. The name will be used later when assigning radiative properties to volume conditions. Determining which set to apply to each volume condition will be easier if you use material names for the sets. The total number of bands was set on the MO > Rad tab, as well as the range of band wavelengths.

The spectral refractive index value for Gray properties will be the same for all wavelengths; that is, Band 1 = 0 - ∞μm. However, if you are using the Non-Gray option, the spectral refractive index value may be different for each

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wavelength band. You set the total number of bands on the MO > Rad tab, as well as the range of band wavelengths.

Insert adds a new row above the active row. Append adds a new row to the end of the table. Delete removes the highlighted row. The first two rows of the table may not be manually altered.

Spectral Refractive Index Sets Dialog Box

SPECULARITY SETS

Specularity data is only required for DOM method with CAFVM option.

The Number of sets box lets you specify the total number of specularity sets. For each specularity set, you may give a name and value. The name will be used later when assigning radiative properties to boundary surfaces. Determining which set to apply to each boundary surface will be easier if you use material names for the sets. The total number of bands was set on the MO > Rad tab, as well as the range of band wavelengths.

Insert adds a new row above the active row. Append adds a new row to the end of the table. Delete removes the highlighted row. The first three rows of the table may not be manually altered.

Specularity Sets Dialog Box


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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 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.


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, that is, 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 may be sufficient. Typically, out experience has shown that a frequency of 100-200 iterations works best for steady

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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 set up and input these properties. The file must be named <modelname>.PATCH, and must reside in the working directory. Its contents look as follows: Line 1: [Name of surface property]

Line 2: [Name of surface substrate material]

Line 3: [Reflection Characteristic]. Options: SPECULAR, DIFFUSE or PARTIALLY_SPECULAR [Specularity] if [Reflection Characteristic] is equal PARTIALLY_SPECULAR

Line 4: [Spectral Characteristic]. Options: GRAY or NONGRAY [Surface emissivity] if [Spectral Characteristic] is equal GRAY

Line 5: [Transparency characteristics] if [Spectracl Characteristic] is equal NONGRAY. Options: OPAQUE or SEMI

Line 6: [Coating Characteristics] if [Spectracl Characteristic] is equal NONGRAY. Options: COATED or UNCOATED

Line 7: [Number of layers in coating] if [Coating Characteristics] is equal COATED

Line 8: [Material name and material thickness] if [Coating Characteristics] is equal COATED and [Surface reflection characteristic] is equal PARTIALLY_SPECULAR

In general, the following guidelines are to be followed while setting up a surface property type (or patch type). Line 1: Name of surface property. 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 CFD's 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 either a real number, representing the value of surface emissivity or the key word USER_SUB to indicate that the emissivity is calculate in user subroutine uemissivity_bc (for example, to calculate a temperature dependent 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


270

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, for example, 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. Example for a patch file:

LAMP

DUMMY

DIFFUSE

GRAY

0.9

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

CHAMBER

DUMMY

DIFFUSE

GRAY

0.1

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

WAFER

DUMMY

DIFFUSE

GRAY

USER_SUB

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

Wafer_top

WAFER

PARTIALLY_SPECULAR

0.1

NONGRAY

ST

UNCOATED

What is the Optical Database File?

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:

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= 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

n0, n1, n2, n3 = coefficients for curve-fit

q = non-dimensional temperature, (T - 300)/1000

(6-36) where

k = imaginary part of refractive index

k0, k1, k2, k3 = 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

ηi = central wave number of i-th band (m-1)

C = 94.40608

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)


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When is P1 Radiation appropriate?

• The P1 model is invalid for non-participating medium. Mathematically, if β 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.

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• 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, β, 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.

What is contained in Radiation Summary tables?

The Radiation Summary Table in CFD-ACE+ provides for each Boundary Record computed data on surface emission, absorption, and net radiation to a surface. These quantities are explained below.

1. Surface Emission And Absorption

At any emitting surface, the total emissive radiative flux qemiss can be expressed in the general non-gray case as

where the summation is over the spectral bands is the number of spectral bands, is the spectral surface emissivity, nl is the spectral refractive index, Fl is the blackbody emissive fraction in the l th spectral band, σ is the Stefan-Boltzmann constant, and Ts is the surface temperature. Note that the STS model in CFD-ACE+ can handle only gray radiation problems and, consequently, L=1 and F=1. Also, n is assumed to be unity in emission computation in all but the CAFVM approach.

If qinc is the incident normal radiative flux on a surface then the reflected radiative flux qref can be written as

where ρl is the reflectivity of the surface.

The part of the incident flux that is not reflected is absorbed by the surface and, thus, the absorbed radiative flux can be expressed as:

At any surface, transmissivity τl, absorptivity αl , and reflectivity ρl sum to unity. In particular, at an opaque surface, transmissivity is zero and, thus, . Also, using Kirchhoff’s law, , which implies .

The net normal radiative flux to the surface, with the normal oriented outward to the medium from which the radiation is incident on the surface, can now be expressed for an opaque non-gray surface as


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If is positive, then it means that the surface absorbs more energy than it emits. The following need to be noted:

• as given above is also valid at semitransparent interfaces in the Monte Carlo method which models the effect of absorption in the medium using surface emissivity.

• At any surface where emissivity is zero, is zero at that surface. This is particularly true at a symmetry surface which is modeled as a purely specularly reflecting surface. In the case of SnDOM and CAFVM, it is also true at a conjugate material interface if the adjacent media are non-opaque, that is, have absorption coefficients greater than or equal to zero, because in these methods it is assumed that there is no absorption and emission at such surfaces.

The columns EMIT, ABSORB and Rad Net in the Radiation Summary table provide area

integrated values of , for each Boundary Record.

2. Volumetric Radiation Source

The Radiative Transport Equation can be integrated over all directions, over the entire spectrum, and over all the boundary surfaces and interfaces to provide the following overall balance equation for radiative flux.

where Sr denotes the volumetric radiation source, ϑ is the total domain volume, NBR is the number of Boundary Records, NBFi is the number of Boundary Faces of the ith Boundary Record, and Af is the area of a boundary cell face. Note that Sr is a function of the absorption coefficient of the medium and is nonzero only in the case of SnDOM and CAFVM approaches as these are the only ones that can handle participating volumetrically absorbing media.

At convergence, the sum total of the values in the Rad Net column, which denotes the net radiative efflux from the system, should be equal to zero in the case of non-participating radiative methods (STS, Monte Carlo), and should be equal to negative of the total energy absorbed by the media in the case of participating radiative methods (SnDOM, CAFVM).

3. Coupled Conduction-Radiation Boundary Conditions

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• At non-isothermal emitting surface, conduction and radiation fluxes are coupled. Energy balances at such surface requires that the algebraic sum of conduction and net radiation fluxes to the surface or interface should be equal to zero.

• At an external system surface with specified heat flux, for example, the algebraic sum of conduction and net radiation flux away from the surface should equal the specified heat flux input to the system.

• At an opaque interface, the algebraic sum of conduction and net radiation flux to the interface should be equal to the conduction flux away from the interface.

References

Azzam, R. and N. Bashara. 1977. Ellipsometry and Polarized Light. New York: Elsevier Noth-Holland.

Chai, J., H.S. Lee, and S.V. Patankar. 1994. "Finite volume method for radiation heat transfer." Journal of Thermophysics and Heat Transfer. 8:3:419-525.

Chang, S. L. and K. T. Rhee. 1984. "Blackbody radiation functions." Int. Commun. Heat Transfer 11:451-455.

Fiveland, W.A. 1988. "Three dimensional radiative heat-transfer solutions by the discrete ordinates method." Journal of Thermophysics and Heat Transfer 2.4: 209-316.

Mathury, J. and S. Mathur. 2000. "Numerical Methods in Heat, Mass, and Momentum Transfer." Draft Notes for ME 608. School of Mechanical Engineering, Purdue University. URL https://engineering.purdue.edu/ME608/webpage/main.pdf

Mazunder, S. and A. Kersch. 2000. "A fast Monte Carlo scheme for thermal radiation in semiconductor processing applications." Numerical Heat Transfer 37.B: 185-199.

Modest, M.F. 1993. Radiative Heat Transfer. New York, Academic Press.

Palik, E. 1985. Handbook of Optical Constants of Solids. New York: Academic Press.

Raithby, G and E.H. Chui. 1990. "A finite-volume method for predicting a radiant heat transfer in enclosures with participating media." Journal of Heat Transfer. 112:2:415-423.

Siegel R. and J. R. Howell. 1981. Thermal Radiation Heat Transfer. 2nd ed. New York: Hemisphere Publishing Corp.

Tan A., D. Wang, K. Srinivasan, and A. J. Przekwas. 1988. "Numerical simulation of coupled radiation and convection for complex geometries." AIAA: 98-2677.

Tan, Z. and J.R. Howell. 1990. "New numerical method for radiation heat transfer in nonhomogeneous participating media." AIAA J. Thermophys Heat Transfer: 4:419-424.

Vaidya, N. 1998. "Multi-dimensional simulation of radiation using an unstructured finite volume method." AIAA Paper No. 98-0857. 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada.


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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:

Applications Features Theory Limitations Implementation Frequently Asked Questions References

Cavitation-Applications

APPLICATIONS

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:

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• 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: Automotive/Hydraulic Applications Turbomachinery Type Problems Hydrofoil Type Problems

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


278

Vane Oil Pump (2000 RPM) - Pressure Profiles Without Cavitation Model (left) and With Cavitation Model (right)

Effect of Cavitation on Volume Flow Rate

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The model has also been used to solve for cavitating flow in a gerotor oil pump as shown below.

Cavitation Inside Pumping Pockets (5000 RPM) of a Gerotor

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.


280

Axial Water Pump Results

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Centrifugal Pumps

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.


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2D Hydrofoil Simulation

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 ρmax/ρmin

Buoyant Flows ~ 1

Transonic Flows ~ 2

Supersonic Flows ~ 10

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Reacting Flows ~ 20

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.

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.


284

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-Theory

THEORY

When a liquid flows into a region where its pressure is reduced to vapor pressure, it boils and vapor pockets develop in it. The vapor bubbles are carried by the flow field until they reach a region of higher pressure, where they suddenly collapse. This process is called cavitation. If the vapor bubbles are near to a solid boundary when they collapse, the forces exerted by the fluid rushing into the cavities create very high-localized pressures that cause pitting of the solid surface. The phenomenon is accompanied by noise and vibrations that have been described as similar to gravel going through a centrifugal pump.

The purity of the liquid in question and the amount of dissolved gases were found to influence the cavitation process. For instance, bubbles in aerated water might withstand several pressure oscillations. This was not observed for pure water.

Tests made on chemically pure liquids show that they would sustain high tensile stresses of the order of mega Pascals. This is in contradiction to the concept of cavities forming when pressure is reduced to the vapor pressure. It is hence generally accepted that cavitation is related to nuclei that enhance bubble growth in low-pressure regions. The nature of nuclei is not thoroughly understood yet.

There are two categories of cavitation: acoustic cavitation and hydrodynamic cavitation

Acoustic cavitation - Pressure waves travelling through a liquid at the speed of sound might cause large pressure fluctuations, which might cause the liquid to boil and evaporate as indicated above. The compressibility of the liquid, the change of liquid and gas properties with pressure and gas volume fraction as well as the speed of sound all influence the cavitation process

Hydrodynamic cavitation - Mainly occurring due to high speed turbulent flow detaching from the surface and the related pressure reductions

Cavitation generally causes several problems, such as:

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• Reduction of useful channel space for liquid flow

• Load asymmetry

• Vibration and noise

• Reduction of machine life

There are also some desirable applications, such as:

• Washing machines

• Surgical procedures

• Liquid-solid separators

• Removal of organic contaminants from water

• Ultrasonic cleaning

Protection against cavitation or enhancing it when required should start with the system design. Simulation techniques resolving the flow conditions in detail could provide a reliable means to predict the susceptibility of the system to cavitate and support optimization efforts.

Simulation models are confronted by two types of challenges:

• Realistic modelling of several interdependent physical phenomena

• Robust numerical procedure for handling inherently steep variations in fluid density, due to very large ratios of liquid, vapor, and gas densities, in conjunction with complex geometries, often with moving parts

To meet such stiff requirements, the present authors have developed and reported the full cavitation model, with implementation in the commercial code, CFD-ACE+. This model has been found to be quite successful, i.e. capable of realistic predictions, without having to adjust empirical coefficients, for a wide range of problems including automotive oil and water pumps, fuel injection systems, high performance rocket turbo-machinery, and biomedical devices with high-frequency piezoelectric motion.

The full cavitation model accounts for all of the first-order physical effects, including liquid-vapor changes, turbulence, surface tension, presence of noncondensible gases, thermal effects and liquid compressibility.

FLOW FIELD AND TURBULENCE

The basic approach consists of using the standard viscous flow equations for variable fluid density and a conventional turbulence model like k - ε.

The fluid density is a function of the vapour mass fraction, which is computed by solving a transport equation coupled with the mass and momentum equations.

(7-1)


286

The vapor volume fraction can be deduced from the vapour mass fraction according to the following equation:

(7-2)

The vapour mass fraction is governed by a transport equation:

(7-3)

The source terms Re and Rc denote vapor generation and condensation and are functions of the flow parameters (pressure, characteristic velocity) and fluid properties (liquid and vapor phase densities, saturation pressure and liquid-vapor surface tension).

The above formulation is derived based on a homogeneous flow approach. This is acceptable for cavitating flows because:

• Cavitation regions often correspond to regions of relatively high velocities. The slip velocity is therefore rather small.

• There are no reliable models describing the local bubble sizes and interface drag forces. Hence, the required overhead to compute a slip velocity cannot be justified.

BUBBLE DYNAMICS

Assuming there are enough nuclei for the inception of cavitation. Based on a zero slip velocity the bubble dynamics can be described by the generalized Rayleigh-Plesset equation.

(7-4)

This equation can be considered to be an equation for void propagation, and hence mixture density.

To obtain an expression of the net phase change rate, the two-phase continuity equations are formulated:

Liquid phase:

(7-5)

Vapor phase:

(7-6)

Mixture:

(7-7)

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Combining the liquid phase, vapor phase, and mixture equations yields:

(7-8)

The vapor volume fraction can be related to the bubble radius by using the bubble number density n:

(7-9)

Substituting equation 6 into 5:

(7-10)

Using the Rayleigh-Plesset equation and neglecting the viscous damping and the surface tension (2nd and 3rd terms on right hand side) and combining equations 2, 3, 5, and 7:

(7-11)

Assuming that initial bubble acceleration effects are negligible the second order derivative of can be eliminated. Using equation 3 from the Flow Field and Turbulence theory chapter and equation 8 we obtain the following equation for vapor transport:

(7-12)

The right hand side of this equation represents the vapor generation (evaporation) rate.

Though it is expected that the bubble collapse process is to be different from that of bubble growth, as a first approximation, this equation is also used to model the collapse (condensation), when p > pB , by using the absolute value of the pressure difference and treating the right hand side as a sink term.

The local far-field pressure is taken to be the same as the cell centre pressure. The bubble pressure is equal to the saturation vapor pressure in absence of dissolved gas, mass transport and viscous damping.

PHASE CHANGE RATES

In equation 9, all terms except n are either known or depended variables. In the absence of a general model for estimating the bubble number density, the phase change rate is rewritten in term of the bubble radius:


288

(7-13)

For simplicity the typical bubble radius is taken to be the same as the limiting (maximum) bubble size. Then the bubble radius can be determined by the balance between aerodynamic drag and surface tension forces. A commonly used correlation is:

(7-14)

For bubble flow regime, Vrel is generally 5-10% of the liquid velocity. By using various limiting arguments, e.g. as , and the fact that the per unit volume phase change rates should be proportional to the

volume fractions of the donor phase, the following expressions for vapor generation / condensation rates are obtained:

(7-15)

(7-16)

Ce and Cc are empirical constants. They were determined by performing several series of computations for sharp-edged orifice and hydrofoil flows. Both of these flows have excellent data, covering a wide range of operating conditions. The most satisfactory values were found to be Ce= 0.02 and Cc = 0.01.

Since this model was implemented it has been used for a large range of technical applications yielding reliable results. Therefore, the present set of values seems quite adequate for general use.

TURBULENCE

The influence of turbulence is treated in a simplified manner. The phase change threshold pressure is raised by the turbulent pressure fluctuations:

(7-17)

(7-18)

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NON-CONDENSIBLE GASES

It is assumed the working fluid is a homogenous mixture of liquid, vapor, and noncondensible gases (NCG), i.e. all three fluids are assumed to be in both mechanical and thermal equilibrium (equal velocity and equal temperature) at each location in the calculation domain. A convection-diffusion equation governs the vapor mass fraction, fv, given by:

(7-19)

The rate expressions were derived from the Rayleigh-Plesset equations, and limiting bubble size considerations. These were expressed as functions of the local pressure, density, turbulence, surface tension, and two empirical constants (Ce and Cc).

The presence of noncondensible gas is accounted for by assuming it to be in a premixed state. Accordingly, the effect of a prescribed uniform constant mass fraction is accounted for through the mixture density equation:

(7-20)

Densities ρv and ρl are functions of saturation pressure and temperature, which are constant for isothermal flows. The gas density ρg is calculated as a function of local pressure, by using the ideal gas law.

The volume fractions are modified as follows:

(7-21)

Taking the noncondensible gas into account and using the turbulent kinetic energy to determine the characteristic flow velocity, equations 3 and 4 in the Phase Change Rates chapter are rewritten:

(7-22)

(7-23)

VARIABLE NONCONDENSIBLE GAS (NCG) FRACTION

In liquid machinery, the noncondensible gas (NCG) exists in two forms:

• In the dissolved state (also referred to as absorbed state) in the liquid, and

• In the evolved or free gaseous state (also referred to as "deaerated" state)


290

Liquids usually contain certain amount of dissolved NCG. The free state NCG can come from premixing of gas in supply liquid, or from local injection at a boundary or by evolution of the dissolved NCG. Thus, to account for full effects of the NCG, we need to track the variation of both free and dissolved gas mass fractions. Furthermore, the process of desorption (evolution, deaeration or release) and absorption (or dissolution) link the two mass fractions and thus need to be accounted for.

We split the NCG mass fraction fg of Equation 2 in the Noncondensible Gases chapter into two components:

(7-24)

It follows from mass conservation that the sum of vapour, liquid, and NCG mass fractions should be unity, or:

(7-25)

Now the mixture density is calculated from:

(7-26)

Where we assume that the density of the liquid with dissolved NCG is the same as that of the pure liquid.

To allow variation and interchange of fg,g and fg,l in the computational domain, we add two more convection-diffusion equations to the original set: one for NCG in gaseous phase, fg.g and one for NCG in dissolved liquid phase, fg.l (note that the mass fraction of the liquid itself is simply fl ).

(7-27)

(7-28)

The total mass of NCGs is conserved in the computational domain, which can be seen by adding the above 2 equations together. The source terms Rd and Ra link the exchange between the two states.

The rate expressions for desorption and absorption of NCG depend on the local partial pressure of NCG in gaseous state, an equilibrium pressure for NCG in dissolved state, and the local mass fraction of NCG. The corresponding expressions are:

(7-29)

(7-30)

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Cd and Ca are empirical coefficients to be determined.

The assumptions in deriving these rate expressions are based on the following considerations:

• Desorption (or release of gas) occurs when the gas partial pressure Pg is below the equilibrium pressure of Pequil . Likewise, absorption occurs when the partial pressure is above the equilibrium pressure. Under both circumstances, the driving force is assumed to be the pressure differential.

Accordingly:

when Pequil > P, desorption rate Rd activates, and Ra is set to zero;

when P > Pequil, absorption rate Ra activates, and Rd is set to zero.

• When the local volume is filled with all gaseous NCG ( fg.g= 1.0) or when the local volume has no dissolved liquid phase NCG ( fg.l = 0 ), the desorption rate Rd is set to zero.

• When the local volume has no "free" NCG ( fg.g= 1.0), the absorption rate Ra is set to zero.

• The value of maximum solubility depends on the fluids (gas and liquid) used and the operating temperatures. At room temperature and pressure, air solubility in water is 2.0e-5, and that of carbon dioxide is 9.6e-4. A value of 0.001 has been chosen for fg,l,lim. For general reference, at 20°C , these values (by volume) are 0.020, 0.96 and 2.5 for air, CO2 and CI2 in water, and the corresponding values at 100°C are 0.012, 0.26, and 0.0.

• The preliminary values of the empirical constants are: Cd = 2.0 and Ca - 0.1. The absorption constant Ca is smaller than desorption constant Cd for two reasons: (1) the absorption process from gaseous state to the dissolved liquid state has to overcome surface tension effect, it experiences much higher resistance; and (2) the pressure differential (Pequil - Pg) during the desorption is bounded by the maximum value of Pequil while it is unbounded during absorption. The absorption process is dependent on the available contact surface area while the desorption process is a volume-process and provided there are sufficient nuclei for bubble formation, will always be faster than the absorption process. The above values were arrived at by parametric studies over a very wide range of flow conditions in orifice problems.

• The equilibrium pressure is assumed to be an elevated saturation pressure to account for turbulent fluctuations in pressure. Since at saturation pressure gas solubility is zero, the saturation pressure is the lowest possible equilibrium pressure.

• The temperature dependence of the absorption/desorption process can be incorporated in the above equations, by allowing the equilibrium pressure to be a function of temperature as well. The functional form of this dependence will depend on the specific liquid and NCG pair used.

Boundary conditions for the above equations will include specification of gaseous and dissolved mass fractions at inlet boundaries, as well as possible inlet conditions for gaseous phase mass fraction fg.g , if needed at a wall injection.

THERMAL EFFECTS

The current original cavitation model assumed a constant temperature flow, and hence constant saturation properties. In a number of engineering problems, external heat transfer in localized regions can lower or increase the local fluid temperatures, and change the saturation properties. This change in turn will affect the cavitation


292

behavior of the flow. Note here that the present formulation is not an attempt to solve problems where a very high concentrated heat input to the fluid flashes the liquid to vapor (e.g. in an ink jet) or boiling phenomena due to high heat addition at a surface. Rather, it is assumed still that the flow cavitates locally due to a combination of both lowering of static pressure and increase in the saturation pressure as a result of thermal changes in the fluid. One of the assumptions mad here is that the liquid and gas phases are in equilibrium, and hence the process of vapour generation is slow, which precludes the treatment of very concentrated heat sources in the volume or surface which can lead to boiling.

Let us assume we have three phases of fluids: liquid, l; vapor, v; and noncondensible gas, g. The total mixture enthalpy is defined as:

(7-31)

and the mixture density is defined as:

(7-32)

or:

(7-33)

The total mixture enthalpy equation satisfies:

(7-34)

Let us express ρh in terms of summation of each individual phase:

(7-35)

If we further express:

(7-36)

And define mixture specific heat as:

(7-37)

We have:

(7-38)

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In the previous expression, hvo − hlo is the latent heat during the phase change from liquid to vapor.

At this point, hlo is the reference enthalpy of liquid and we can set it to a reference value of zero. Since there is no latent heat involved for the noncondensible gas, we will set hgo , as well.

Now, equation 4 becomes

(7-39)

where the underlined term is actually

(7-40)

with Re as the vapor generation rate, and Rc as the vapor condensation rate.

LIQUID COMPRESSIBILITY

The definition for mixture density with compressible liquid has the same format as before:

(7-41)

with the exception that now the liquid density ρl is taken as a function of pressure (and temperature as needed), instead of a constant.

In the compressible fluid treatment, we need to account for the fluid compressibility ∂ρ/∂p. This is calculated by taking the derivative of equation 1

(7-42)

or

(7-43)

the first term represents the compressibility of vapor phase, the second term is the compressibility of gaseous phase in the liquid and the last term represents the liquid compressibility.

Universal liquid compressibility relations linking the liquid density with pressure and temperature are difficult to obtain for liquids in general. A simple approach is to use the liquid bulk modulus κ defined as:


294

(7-44)

The liquid bulk modulus can vary with temperature. Thus, the variations in bulk modulus can be easily incorporated in the fluid properties. Since the bulk modulus does not depend on pressure level, we can integrate equation 4 to obtain a relation between the density and pressure for the liquid, at a given temperature:

(7-45)

The liquid reference density and pressure can be defined at room temperature. For water, these values can be set to Pc

=1atm

and ρκ

= 103 kg/m3

.

In some cases, the liquid compressibility is specified simply in terms of the speed of sound in the liquid. In this case, we can use the definition of the speed of sound, C

(7-46)

to calculate the value of the bulk modulus as:

(7-47)

As seen in equation 3, the total compressibility of the mixture consists of the compressibility of vapor, NCG, and liquid. The vapor is assumed to be incompressible as vapor pressure is assumed to be at saturation pressure all the time. For NCG, the idea gas law is used.

Cavitation-Implementation

IMPLEMENTATION

The Implementation section describes how to set up a model for simulation using the Cavitation 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 Cavitation Module related inputs to the CFD-ACE-Solver

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

GRID GENERATION

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.


CFD-ACE+ provides the inputs required for the Cavitation Module. Model setup and solution requires data for the following panels:


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.

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 Cavitation Module are located under the Cavitation (Cav) tab. The Model Options section includes:

Shared Cavitation (Cav) Liquid Non-Condensable Gas Concentration Phase Change Coefficients


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SHARED

There are no settings under the Shared tab affect the Cavitation Module. (See Control Panel-Model Options for details.)

CAVITATION (CAV)

All of the model options for the Cavitation Module are located under the Cavitation (Cav)).

Model Options Panel in Cavitation Module Mode

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.

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 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

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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.

PHASE CHANGE COEFFICIENTS

The phase change rate coefficients (Ce and Cc) can be specified here. These coefficients are used as described in equation 7-22 and equation 7-23. 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.

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

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.


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These properties should be evaluated at the operating temperature which was specified in the Cavitation Model Options area (see Model Options Settings).

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.

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.

SOLVER CONTROL

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:

Spatial Differencing Scheme Solver Selection Under Relaxation Parameters Variables Limits


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.

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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.


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.


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.

OUTPUT

There are no settings under the Output tab that affect the Cavitation Module. See Control Panel-Output Options for details.

SUMMARY OUTPUT

There are no settings under the Summaries tab that effect the Cavitation Module. See Control Panel-Summary Output for details on the general printed output options including boundary integral output, diagnostics and monitor point output.

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:


300

Density Ratios in Engineering Flows Variable Units

Total Void Fraction (α = αv + αg) ---

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 Variable Descripton Units

MassFr Vapor Mass fraction ---

Total_Volume_Fraction Total volume fraction (Void

Fraction) ---

Vapor_Volume_Fraction Vapor volume fraction ---


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 7-22 and equation 7-23). 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.

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References

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

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:

Applications Features Limitations Implementation Frequently Asked Questions


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Applications

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:

Fluid-Structures Interaction Problems Simple Prescribed Motion User Defined Motion

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.

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.

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.

Features and Limitations

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:

Automatic Remeshing User Defined Remeshing

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.

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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.

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 μ and λ are the Lame constants, expressed in terms of material properties as:

(8-2)

(8-3)

In equations 8-2 and 8-3, E is the modulus of elasticity and γ is Poisson’s ratio. For this application, γ 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.


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Finite Element Solution

Equation 8-4 is solved using a standard Galerkin formulation [2], which can be expressed as

(8-5)

Where Φαis the shape function for node α 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 β. 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

• This scheme applies to both structured and unstructured grids (Tetrahedrals, Prisms, and Pyramids).

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• 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.

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.

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.


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Implementation

IMPLEMENTATION

The Implementation section describes how to setup a model for simulation with the Grid Deformation Module. The Grid Deformation Implementation section includes:

Grid Generation - Describes the types of grids that are allowed and general grid guidelines 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. Post Processing - Provides tips on what to look for in the solution output

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.


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, AND Boundary Conditions.

PROBLEM TYPE

Click the Problem Type [PT] tab to see the Problem Type Panel. See Control Panel-Problem Type for details. Select Grid Deformation to activate the Grid Deformation Module. This module can work with any of the other CFD-ACE+ modules.

MODEL OPTIONS

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. The Model Options section includes: Shared, Deform, Auto Remesh, and User Subroutine.

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.)

DEFORM

All of the model options for the Grid Deformation Module are located under the Grid Deformation (Deform) tab.

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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.

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-UGRID for details.

VOLUME CONDITIONS

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.

BOUNDARY CONDITIONS

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).


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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.

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

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.

Variable Description

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 Ange

Rotation Angle Rotation angle in degrees (for Rotation Angle mode only).

Rotation Omega Rotation annular velocity in degree/second (for Angular Velocity mode only)

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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.

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.

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.


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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.

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.

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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.

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.


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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.

Figure 7 - Moving Edge ae. Point e is a corner points of domain aee’c.

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Figure 8 - Final Grid After Dividing Domain

Figure 9 shows the multi domain structure.

Figure 9 - Multi Domain Structures

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.


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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.

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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.


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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.

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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.

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.


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

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.


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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.

Applications

APPLICATIONS OVERVIEW

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:

Pure Structural Analysis Coupled Solid/Fluid/Thermal Problems Multidisciplinary Electrostatic Problems-MEMS 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.

Thermoelastic Analysis of a Gas Turbine Atomizer

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.

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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

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).


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Coupled Fluid/Thermal/Structural Analysis of an Orifice

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 depicts a model of an accelerator, which is an electrostatic loaded plate clamped by four beams. This plate sits 2µm 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.

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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 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.


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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.

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Solid Model of a Mesopump Cell

Electrostatic Field Distribution in a Pump


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Features

The Stress Module supports the following structural analysis capabilities:

• Steady-state and transient analysis • Linear analysis • 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 • Limitations • The Stress Module has the following limitations:

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.

THIN WALL BOUNDARY CONDITIONS

Stress calculations cannot be made for a thin wall boundary condition.

LIMITED ELEMENT LIBRARY

The elements supported by FEMSTRESS include 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.

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Theory

THEORY OVERVIEW

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 ε 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, ε0 and σ0 are initial strains and stresses, respectively.

Thermoelastic stress problems are handled by considering the temperature rise T to contribute to initial strains as:

(9-5)

where αi 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 δa, the work done by the external and internal forces, respectively, are

(9-6)


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(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. At each iteration we solve for a correction to the current displacement field using:

(9-9)

The rate of change of Ψ 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,

(9-13)

where:

(9-14)

The integral in equation 9-13 may be written as (Zienkiewicz, 1971)

(9-15)

where Kσ is known as the initial stress matrix or geometric matrix.

Thus, the equation solved in the Newton-Raphson scheme is

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(9-16)

where:

(9-17)

Convergence is obtained when the maximum correction Δa 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.

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).


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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 α and β are the mass and stiffness proportional damping coefficients respectively. With this formulation, the critical damping fraction, as a function of frequency, may be expressed as:

(9-19)

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The two damping coefficients α and β are obtained by specifying fractions of critical damping ( ξ1 and ξ2 ) at two frequencies (ω1 and ω2). This yields two equations in two unknowns which may be solved as:

(9-20)

(9-21)

If the values of α and β are known, they may be entered directly in CFD-ACE+. Otherwise the four values, ξ1, ω1, ξ2,and ω2 may be specified, and α and β will be calculated internally.

The frequency values ω1 and ω2 are usually chosen to bound the design spectrum of the problem. In such a case, ω1 is taken as the lowest natural frequency of the structure (which may be obtained from a modal analysis) and ω2 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 αM decreases with increasing frequency, whereas the βK 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 ω1 and ω2 with critical damping fractions ξ1 and ξ2 respectively. This figure demonstrates why ω1 and ω2 are chosen to bound the design spectrum, as the amount of damping increases substantially outside this range.


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Fraction of Critical Damping versus Frequency for Raleigh Damping. Contribution of Stiffness and Mass Proportional Damping to Total Damping Included


What is the Cartesian stress tensor?

At each point, we have the stress tensor σij, where i refers to the face and j refers to the direction:

The nine components σij make up the Cartesian stress tensor. The stress tensor is symmetric, that is, σij= σji, 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,

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(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 σmax Planes where is minimum give σmin The maximum shear stress is given by

(9-30)

where τmax acts on planes bisecting the planes of σmax and σmin. The VonMises stress is related to the distortional energy of the body (as opposed to the hydrostatic) and is given by

(9-31)

where σ1, σ2, and σ3 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?

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?


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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 • Tetrahedral, Prismatic, Pyramidal • Standard Brick Elements (First and Second Order) • 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.

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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: • 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.


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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

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

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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

Examples

The Oil Flow through a Compliant Orifice Tutorial uses the Stress Module with one or more other Modules.

Stress Concentration Demo

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.

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.


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Stress Concentration Geometry

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).

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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, ν=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.

Hoop Stress Demo

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 • ν = 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.


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Radial Displacement Contours

Large Deflection Demo

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 • ν = 0.0, • ρ = 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).

Large Deflection of a Cantilevered Beam

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Normalized Tip Deflections

h/L V/L

Analytical 0.445 0.830

CFD-ACE-GUI 0.443 0.832

Stress Concentration in a Circular Cylinder Validation Case

REFERENCE Roark and Young, Formulas for Stress and Strain, Page 600.

ELEMENTS 4 noded tetrahedrals

PROPERTIES E=200 Gpa, ν=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.


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Stress Concentration in a Circular Cylinder

RESULTS Maximum stress concentration: Roark & Young: 1.33 Stress Module: 1.33

Thermoelastic Deformation of a Cylinder Validation Case

REFERENCE Boley and Weiner, Theory of Thermal Stresses, pg 290.

ELEMENTS 4-noded quadrilaterals in axisymmetry.

PROPERTIES E=20 Gpa, ν = 0.0, α = 5.0e-5 K-1

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DETAILS

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 ν=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 ν=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) σinner(Mpa) σouter(Mpa)

Analytical 0.00317 0.0159 -73.10 26.90

CFD-ACE-GUI 0.00318 0.0159 -68.78 26.62


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Thermoelastic Analysis in a Cylinder

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.

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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.

Vu-Quoc, L and Tan, X.G. "Optimal solid shells for nonlinear analyses of multilayer composites. Part I: Statics." Computer Methods in Applied Mechanics and Engineering, 2003; Vol.192, pp. 975-1016, .

Vu-Quoc, L and Tan, X.G. "Optimal solid shells for nonlinear analyses of multilayer composites. Part II: Dynamics", Computer Methods in Applied Mechanics and Engineering, 2003. Vol.192, pp. 1017-1059.

Zienkiewicz, O.C., The Finite Element Method in Engineering Science. McGraw-Hill, 1971.

Electric Module

Introduction

The electric module solves for the following governing equations for electric potential distribution:

• Electrostatic Equation in dielectrics • Electric Conduction equation for currents in conductors and loss dielectric. This includes DC conduction,

AC conduction (sinusoidal steady state) and Transient Electrical Conduction.

The electric field E, the capacitance, the electrostatic pressure forces, and the conduction currents are calculated from the electric potential.

Two different techniques are available for solving Poisson’s equation: Finite Volume Method (FVM), and Boundary Element Method (BEM). The solution methods account for variation in relative permittivity, space charge, and electrical conductivity in the computational domain. The Electric Module section includes:

Applications Features Limitations Theory Implementation Symbol Definitions References

Applications

Common uses of the Electric Module are given below. This is not a complete list and many more uses are possible.

• Capacitance of metal/dielectric structures such as parallel plates, concentric spheres, etc.


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• Electric fields due to distributions of space charge and equipotential contacts. • Electrostatic loading of mechanical structures. The Electric Module calculates the electrostatic pressure

force which can be used with the Stress Module to determine the structural response to electrostatic loads. A prime example of this would be micro systems such as a doubly clamped beams, accelerometers, high frequency resonators, electrostatic torsional micro mirrors, linear lateral resonator comb drives, angular resonator comb drives, and fluid damped beams.

• Electrostatic force on a conductive bath. In conjunction with the Free Surfaces (VOF) Module, the Electric Module can calculate the force on a conductive fluid, addressing problems such as the extraction of a conductive fluid from a bath.

• Joule heating. • Current density as a source for a magnetic field calculation. • Electrical field distribution in an electro-kinetically driven flow field as it occur some biochips and

bioMEMS devices. • Simulation of nonuniform electric field (AC field) in a Dielectrophoretic (DEP) system. • Electric field distribution in the electro-fluidic system to calculate pH value. • Electric Module -

Features You can use the Electric Module as stand-alone or with other modules for multi-disciplinary design, e.g., electromechanical, electrothermal, electrochemical, electrofluidic, etc. The Electric Module has several unique features:

• A multi-pole accelerated BEM solver which has a solution speed on the order of O(N) fast where N is the number of faces and does not require volume meshing;

• An FVM solver that can effectively solve complex multi-physics and nonlinear problems • Support both structured and unstructured meshes (tetrahedral, hexahedral, prisms, and general

polyhedrals) • Coupling with other physics such as the Heat Transfer, Stress, Free Surfaces (VOF), and Magnetic Modules. • Coupling with dynamic problems such as moving deforming materials and moving space charges (e.g.,

charged particles).

Limitations Specifically with a Plasma/CCP model, if Parametric setting is chosen, there can be only one continuous boundary edge (2D or axisymmetric model) or surface (3D model) assigned as Parametric.

Theory

THEORY

Topics in this section include the following:

Electrostatic Fields in Dielectrics Electric Conduction (Currents in Conductors and Lossy Dielectrics) Leakage Currents FVM and BEM Electrostatics Solution Techniques

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Calculating Auxiliary Quantities that Depend on the Electric Potential Peltier/Seebeck Effect

ELECTROSTATIC FIELDS IN DIELECTRICS

Electrostatic fields are generated by a combination of stationary charges and applied potentials. Electrostatics is governed by Gauss’ Law, which states that the net electric flux passing through a closed surface is equal to the net charge enclosed by that surface. In differential form and mks units, Gauss’s Law is written as

(10-1)

where the vector D is the electric displacement flux density (C/m2) and ρ is the volume charge density, (C/m3). The electric flux density, D, and electric field, E, are related by the constitutive relation:

(10-2)

where:

(10-3)

In electrostatics, the electric field vector, E, is irrotational (i.e., the curl of the electric field is zero, ) so a scalar electric potential, φ, can be defined as:

(10-4)

Inserting equation 10-2, equation 10-3, and equation 10-4 into equation 10-1 we get Poisson’s equation for the electric potential:

(10-5)

Equation 10-5 is solved by the electrostatic option of the Electric Module if the Electrostatics option is selected. After the electric potential φ is solved using equation 10-5, CFD-ACE+ computes other quantities that depend on φ. These other quantities include the electric field, surface charge, capacitance, electrostatic pressure force, energy contained in the electric field, and the electric virtual force (energy gradients).

Electric fields are calculated in all dielectric materials and in free space surrounding them. In electrostatics, it is assumed that conductors have zero electric field (φ is constant). To specify that a material is a conductor in the electrostatic option of the Electric Module the electrical conductivity has to be set to Perfect Conductor.

The boundary conditions for φ are:

• at all metal contacts specify the electric potential or voltage • on external boundaries apply zero voltage or surface charge density • at the interfaces between two dielectrics . Where En is the normal electric field at side 1

or 2. Apply the Ignore or Dielectric option.

The Electric Theory section includes:

Electric Conduction (Currents in Conductors and Lossy Dielectrics)


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Leakage Currents FVM and BEM Electrostatics Solutions Calculating Auxiliary Quantities that Depend on Electric Potential Peltier/Seebeck Effect

ELECTRIC CONDUCTION (CURRENTS IN CONDUCTORS AND LOSSY DIELECTRICS)

The Electric Conduction option calculates the current density J in conductors and lossy dielectrics. The fundamental equation governing the current flow is the electrical current continuity equation:

(10-6)

where J is the electric current density (A/m2) and ρ (C/m3) is the charge density. Substituting ρ of Gauss’ Law for the Electric Field ( equation 10-1) into the continuity equation (equation 10-6) and assuming the time derivative and the divergence operators commute results in:

(10-7)

Next, equation 10-7 is written in terms of the electric field. Assuming the total current density J is due only to conduction current (e.g., J = Jc = σE), and all the medium are isotropic (e.g., D = εE) equation 10-7 becomes:

(10-8)

where σ is the electric conductivity measured in (Ω-m)-1 (note 1/σ is the resistivity).

The electrical resistivity 1/σ can be specified as either a constant value or a function of temperature:

(10-9)

where:

σo = electrical conductance at temperature To

a = the temperature coefficient of resistivity.

Finally, equation 10-8 is written in terms of the electric potential. Using the definition of the electric field in terms of the electric potential and the magnetic vector potential

and assuming that there are no time varying magnetic fields (i.e., ) results in which is substituted into equation 10-8.

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(10-10)

The general equation for the electric potential in a conductor or lossy dielectric is given by equation 10-10.

There are three different options (approximations) for solving the conduction problem (equation 10-10): DC conduction, AC conduction (one frequency), and time domain conduction.

After the electric potential φ is solved using equation 10-10 the Electric Module computes other quantities that depend on φ. These other quantities include resistance or impedance, source currents (for the Magnetic Module), joule heating sources (for the Heat Transfer Module) and body forces (for the Free Surfaces (VOF) Module).

DC CONDUCTION

The DC Conduction option assumes steady state conditions ( ). Assuming steady state or DC currents, equation 10-10 becomes a Laplace equation for the electric potential:

(10-11)

Equation 10-11 is solved by the DC conduction option.

AC CONDUCTION (ONE FREQUENCY)

The AC Conduction option solves the current continuity equation assuming a sinusoidal steady state. Converting the current continuity equation ( equation 10-10) into the frequency domain where the electric potential becomes a complex quantity yields:

(10-12)

The electric potential of this equation is complex (i.e., ) so two equation need to be solved:

(10-13)

(10-14)

For many cases the dc conduction problem will suffice. For ω = 0.0 or where , equation 10-12 reduces to the DC conduction problem ( equation 10-11). For a typical conductor σ ≈ 106 Ω-1m-1 and ε ≈ εo = 8.85x10-12 C2/Nm2. So the DC conduction equation is appropriate for frequencies (f = ω/2p) on the order of 1010 Hz or less. In the AC conduction option the real and imaginary part of the electric potential in equation 10-12 are solved.

TRANSIENT ELECTRICAL CONDUCTION (TIME DOMAIN)

To perform transient (time domain) current flows equation 10-15 is solved:

(10-15)

In the transient conduction option of the Electric Module, equation 10-10 is solved for the electric potential φ. To achieve economical solution speed, a variable time step may have to be used.


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LEAKAGE CURRENTS

Imagine a capacitor with two dielectric materials between two contact electrodes connected to a charging circuit (e.g., battery) as shown:

Current Circuit Leakage

If the electric charging circuit charges the metal contact instantaneously an electric field will be established in the dielectric materials (see Figure - B section) but no current (free charges) will flow for some time. For this (charging) problem, use the electrostatic option in the Electric Module with permittivity as the material property.

At a later time when free charges accumulate at the dielectric interface, the leakage current starts to flow (see figure's C diagram). To simulate the steady current flow, use the DC conduction option in the Electric Module with conductivity (resistivity) as the material property. The time constant for the charging process (and leakage currents) depends on the dielectric materials properties (τ = ε/σ).

FVM AND BEM ELECTROSTATICS SOLUTION TECHNIQUES

Two different techniques are available for solving the electrostatic equation:

• FVM (Finite Volume Method) - The FVM requires a volume mesh and calculates the electric potential (and electric field) at every cell (volume) in the mesh. The BEM requires only a surface mesh (although volume information is required in specifying material volume conditions) and calculates the electric potential only at boundary and interface faces.

• BEM (Boundary Element Method) - The BEM method is recommended for open boundary problems, for problems with small surface/volume bodies, external fields, etc. FVM method is recommended for enclosed systems, for large surface/volume bodies, for electrical current flows in conductors, for electro-thermal simulations, etc.

CALCULATING AUXILIARY QUANTITIES THAT DEPEND ON THE ELECTRIC POTENTIAL

SURFACE CHARGE DENSITY AND TOTAL CHARGE Surface charge density, ρs, on an elementary surface element is:

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(10-16)

where n is the unit vector perpendicular to the surface. Total electric charge on an electrode (or conductor) is

(10-17)

where ε is the permittivity of the dielectric that surrounds the conductor.

CAPACITANCE

If an electrical potential is applied between two separated conductor plates, a charge +Q appears on one plate and -Q on the other, and an electric field, E, is established between the plates.

Electric Field Due to a Voltage Difference on the Plates of a Parallel Plate Capacitor

The ratio between the separated charge (Q) and an electrical potential difference is called capacitance, C (C = Q/V). The unit of capacitance is the Farad which is Coulomb/Volt.

Capacitance (Q/V) is a purely geometrical quantity determined by sizes, shapes and separation of conductors. V by definition is the potential of positive conductor less that of negative conductor and Q is the charge of the positive conductor. Capacitance is a positive quantity (C > 0).

In the Electric Module a total charge is calculated for every set of boundaries with a unique fixed electric potential φo. To determine the Q at a fixed potential boundary, one starts with the electric field boundary conditions.


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Defining region numbers and direction of normal for equation 10-18.

(10-18)

(10-19)

Equation 10-18 states that the discontinuity at an interface in the normal component of the displacement flux is equal to the surface charge at the interface. Assume one region (2) is metallic so that the electric field goes to zero there. The total charge on a contact is then given by equation 10-20. The total charge is the sum at each face with a fixed potential (φo) of the incident displacement flux normal to the fixed electric potential boundary (εiEni) times the area of the face (Ai).

(10-20)

A capacitance is calculated for each set of faces with a unique potential boundary condition. First, the total charge on each collection of faces with a unique fixed potential boundary condition given by equation 10-20 is calculated.

Then for those face sets with φ ¹ 0.0 the ratio is calculated.

CAPACITANCE MATRIX FOR MULTIPLE CONDUCTORS

In several applications (e.g., electronic interconnects of integrated circuits) large numbers of conductors are arranged in close proximity to each other. In electronic integrated circuits made of conductive materials (e.g., Al, Cu,...) separated by a dielectric (e.g., SiO2), a large number of capacitances exist. There may be desired capacitance effects or parasitic effects that have unwanted influence on circuit performance. Imagine three conductors and a ground plate (see figure below). If one of the conductors has an applied voltage of unity (e.g., V = 1) and others are grounded (e.g., V = 0). A capacitance matrix can be formed as:

(10-21)

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Capacitances of Three Conductor Lines and a Ground Plate

The charges on the three contacts are calculated by the electrostatic option of the Electric Module. The capacitances can be determined using the three equations in equation 10-21. The mutual capacitances (C12 = -Q2, C13 = -Q3) are equal to the charges induced at the grounded contacts. The self-capacitance is equal to the charge on conductor 1.

(10-22)

The capacitance matrix relates charges Q and voltages V for any number of conductors.

(10-23)

The resistance R12 between any pair of contacts 1 and 2 is calculated using Ohm’s Law:

(10-24)

where I12 is the total current flow in the conductor from contact 1 to 2 and V12 is the electric potential difference between contact 1 and 2. The total current is obtain by summing the dot product of current density with the area elements at the contacts (J²•AnA). The potential difference is obtained by V12 = φ1 - φ2.

ELECTROSTATIC PRESSURE FORCE (ELECTRICAL/STRUCTURAL)

The coupling between the electrostatic option in the Electric Module and the Stress Module is through pressure forces. The Electric Module solves Poisson’s equation and uses the electric fields to calculate the pressure force. The Stress Module uses the pressure forces as boundary conditions.

Electrostatic Pressure (P) is calculated using the following relation (Bachtold, 1997):

(10-25)


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The electrostatic pressure on the conductor surface is always triggering for interfaces between two dielectric materials with different electric permittivity values ε1 and ε2. Electrostatic pressure is given by:

(10-26)

Electrostatic pressure is tugging at the material with larger relative permittivity.

VIRTUAL ELECTROSTATIC (DIELECTRIC) FORCE (ELECTRIC/FREE SURFACES)

A dielectric force arises when an electric field exists in a region with an inhomogeneous electric permittivity distribution. This optional force is currently available when the Electric and Free Surface (VOF) modules are activated, and accounts for the dielectric force due to the difference between the electric permittivities of Fluid 1 and Fluid 2.

The Kevin-Helmholtz virtual work method for force calculation can be applied to the Maxwell stress tensor to yield an expression for the electromagnetic force density (Vágó and Gyimesi). The dielectric force is one of the terms from this force density expression, and is evaluated as

where ε0 is the electric free-space permittivity and εr is the material electric relative permittivity. The direction of this force is given by the direction of the gradient of the relative permittivity.

To couple the Electric Module with the VOF Module, the VOF module is given access to the electric material properties (particularly the electric relative permittivity), and recalculates them as a function of fluid fraction.

Note that the dielectric force is applied only to cells that have a fractional content of Fluid 2; that is, cells where F > 0. Elsewhere in the domain (for cells where F = 0) the dielectric force is not applied. For each cell where the dielectric force is applied, the calculated force is transferred to the fluid as a body force (that is, as a source term to the momentum equation).

ELECTRIC CURRENT DENSITY (ELECTRIC/MAGNETIC)

In the DC conduction option, current sources can be calculated for the Magnetic Module. This is a useful method for describing source currents in that you do not have to specify a direction of the current, it enables the use of source currents of arbitrary geometry, and it allows the current density direction to reflect any changes due to a moving grid.

(10-28)

JOULE HEATING (ELECTRIC/HEAT TRANSFER)

In the DC, as well as the AC, conduction option of the Electric Module Joule heating sources are calculated when it is coupled with the Heat Transfer Module. The heat generated in a conductor due to current flow through conductive materials (Joule heating) is:

(10-29)

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PELTIER/SEEBECK EFFECT

The thermoelectric Seebeck effect generates electrical potential difference in any conducting material with temperature gradient. The value of this potential is proportional to temperature dependent material Seebeck coefficient and temperature difference. This effect is not connected with material junction or Joule heating effect.

When the electrical current flows through a material with non-zero Seebeck coefficient two additional effects are observed. The reversible change in heat content at an interface between dissimilar conductors, called Peltier effect, and change of heat content within conductor, called Thomson effect.

SEEBECK EFFECT

Seebeck Effect

The temperature gradient within isolated conductor generates electrical potential. The value of this potential is proportional to Seebeck coefficient of material and the temperature difference. The Seebeck coefficient is temperature dependent.

where:

PELTIER EFFECT

Peltier Effect

The flow of current through materials junction absorbs power proportional to relative Peltier coefficient π21 and current value I. The direction of current flow and a sign of coefficient determine whether heat is liberated or absorbed. The relative coefficient is equal to difference between absolute material Peltier coefficients π21= π2- π1. The Peltier effect is a result of the change in the entropy of the electrical charge carriers as they cross the junction.


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where:

THOMSON EFFECT

Thomson Effect

The flow of current through a material with temperature gradient absorbs power proportional to Thomson coefficient b, the temperature difference, and the current value I.

where:

COEFFICIENT VALUES

The electrical and thermal energy conservation equation for thermoelectric relationship leads to following equation:

The zero net change in the enthalpy approximation leads to another equation:

Those two equations were used to find dependence between Peltier, Thomson, and Seebeck coefficients. This leads to following formulas:

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Therefore:

PELTIER/SEEBECK-ACE+ IMPLEMENTATION

HEAT ABSORBED/GENERATED IN THE HEAT MODULE

Both Peltier and Thomson effect create additional heat sources for the Heat module.

The Peltier effect absorbs (generates) heat on the boundary between two materials. Those materials should have different Seebeck coefficients. This effect is implemented as surface heat source. For each face the difference between Seebeck coefficients on both sides is computed. Than the result value is multiplied by face temperature and electrical current flowing through this face. The heat source is added to adjoining cells with weighting factor of distance and thermal conductivity.

Therefore:

The Thomson effect absorbs (generates) heat inside material. The material should have temperature dependent Seebeck coefficient. This effect is implemented as a volume heat source. The equation for this effect was changed in the following way.

where:

DC CONDUCTION IN ELECTRIC MODULE

The Electric module in ACE+ for DC conduction problem solves the Laplace equation:

then it computes the electric field as:


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where:

The introduction of thermoelectric Seebeck effect will add an additional source term to those equations:

where:

S - thermoelectric Seebeck coefficient

The first equation is rearranged to following form:

The right side of this equation is the additional source term for the Electric module. The second equation (Eq. 2) modifies the electric field equation in the same module.

KNOWN PROBLEM

Those two equations should not generate artificial currents after converged solution, since the terms added to first and second equations are the same. Unfortunately, because is computed differently in equations (1) and (2), due to different discretization schemes, those terms do not reduce themselves. This problem generates artificially big value of E at local points and solution instability. The high local current J=σE generates heat and prevents convergence in the Heat module. That effect influents mutually ∇ T term in equations.

PELTIER/SEEBECK-VALIDATION

A 3D simulation of a rectangular test structure was performed.

The electric current flows from the right side of structure to the grounded left side. All other sides were set to adiabatic conditions and were electrically isolated.

The structure has rectangular uniform cross-section.

The current was set to 1A per 1mm2 of cross-section. The structure in x direction is build from three layers, counting from left, 5 mm metal conductor, 5 mm thermoelectric material, and 5 mm of metal conductor.

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The metal property was set to the following values:

• Electrical conductivity 108 1/Ω-m

• Thermal conductivity 100 W/m-K

Thermal conductivity of the thermoelectric material was 1 W/m-K.

PELTIER EFFECT TEST

The Seebeck coefficient of the thermoelectric material was set to 200 µV/K.

For this case the electrical conductivity of the material was set to high value 107 1/Ω-m, to make the Joule effect negligible (PJ = R I2 = 0.5mW) comparing to Peltier effect:

The thermal resistance of the thermoelectric is:

Therefore the temperature drop on the thermoelectric should be approximately:

The images below present the result of ACE+ simulation. The first one presents a temperature map, the second the temperature graph.

Peltier Effect Temperature Map


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Peltier Effect Temperature Graph

PELTIER EFFECT WITH JOULE HEATING

The Seebeck coefficient of the material was 200 µV/K. For this case, the electrical conductivity of the material was 80,000 1/Ω-m. The total Joule heating in this case (PJ = R I2 = 62.5mW) is comparable to the Peltier effect.

The analytical solution for 1D case gives following results:

• Temperature at x=5mm (left side of TE material) should be 304K

• Temperature at x=10mm (right side of TE material) should be 232K

• Maximum temperature in TE should reach 315K at x=6.3mm

The images below present the result of ACE+ simulation. The first one presents temperature map, the second the temperature graph.

Peltier Effect with Joule Heating Temperature Map

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Peltier Effect with Joule Heating Temperature Graph

PELTIER AND THOMSON EFFECTS

The Seebeck coefficient of the material was set to:

For this case the electrical conductivity of material was set to high value 107 1/Ω-m.

Peltier and Thomson Effects Temperature Map


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Peltier and Thomson Effects Temperature Graph

PELTIER/SEEBECK-THERMOELECTRIC EFFECTS EXAMPLE

VERIFICATION CASE 1

Two elongated blocks of thermoelectric materials with different Seebeck coefficients were connected together at one end (right-hand side in Figure 1). The isothermal boundary condition of 400K was set for the connected end (right-hand side in Figure 1) and the isothermal condition of 300K to opposite side of the blocks (left-hand side in Figure 1).

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Figure 1: Temperature Distribution Calculated by ACE+ Heat Module

The Seebeck coefficient of the top block was set to 500 µV/K (tellurium). The coefficient of the bottom block was set to -75 µV/K (bismuth). Then, the bottom block was grounded at its left end (electrical potential = 0V). All the other sides were electrically isolated.

The structure was simulated with Heat Transfer and Electric DC Conduction modules. The Thermoelectric (TE) effects in both modules were turned on. There is no current flow in the circuit, because all sides except one point are isolated. The temperature gradient should generate electrical potential drop inside TE materials. The ΔV = S ΔT, therefore voltage drop in the bottom block should be: 75µV/K·100K=7.5mV. For the top block: ΔV = 500µV/K·100K = 0.05V

This is the theoretical solution. The electric potential obtained from ACE+ simulation is shown in Figure 2. The obtained values accurately correspond to the theoretical ones.


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Figure 2: Electric Potential Distribution Calculated by ACE+ Estat Module

Implementation

IMPLEMENTATION

The Implementation section describes how to setup a model for simulation using the Electric Module.

Grid Generation - Describe the types of grids that are allowed and gridding guidelines Model Setup and Solution - Describes the Electric Module related inputs to the CFD-ACE-Solver Post Processing - Provides tips on what to look for in the solution output

GRID GENERATION

The Electric Module can be applied to any geometric system (3D, 2D planar, or 2D axisymmetric). and supports all grid cell types (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.

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CFD-ACE+ provides the inputs required for the Electric Module. Model setup and solution requires data for the following panels:

Problem Type Model Options Volume Conditions Boundary Conditions Coupling with Other Modules Initial Conditions Solver Control Output

PROBLEM TYPE

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

MODEL OPTIONS

Click the Model Options [MO] tab to see the Model Options Panel. See Control Panel-Model Options for details. The Model Options section includes:

Shared Electric Flow Heat

SHARED

There are no settings under the Shared tab that affect the Electric Module. See Control Panel-Model Options for details.

ELECTRIC

The model options for the Electric Module are located under the Electric (Electr) tab.

The Model Options Electric tab includes:


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Electric Field Options Electric Solver Capacitance Matrix Peltier/Seebeck Effect

Electric Field Options

The Electric Field Options calculate the electric potential and include:

Electrostatics Solves Poisson’s equation.

DC Conduction Solves the DC steady-state conduction equation.

AC Conduction (One Frequency)

Activating this option requires you to select a single frequency for the electric field. When this option is used in a transient problem, a convert to time domain checkbox is available. If convert to time domain is not selected, period averaged values of the electric field will be used as sources in the other modules (for example, Flow, Chemistry). If convert to time domain is activated, instantaneous values will be used instead.

Time Domain Available only when the Transient option is selected in the Shared tab.

Piezoelectric Solves the coupled electromechanical system for piezoelectric materials.

Electric Solver

The Electric Solver only applies to Electrostatics. There are two methods available to solve the equation: Finite Volume Method (FVM) and Boundary Element Method (BEM).

Finite Volume Method (FVM)

When the Finite Volume Method is chosen, the Model Options panel appears. See Theory-FVM for details on finite volume method.

Boundary Element Method (BEM)

When the Boundary Element Method is chosen, the Advance Settings tab in the Solver Control panel appears. The BEM method is a specialized solver and is only valid for computing fields on surfaces with no grid between them.

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Advanced tab in the Solver Control Panel in Electric Module Settings (for BEM Solver) Mode

Unbounded - The Unbounded toggle specifies if the problem is bounded or unbounded. A bounded problem assumes the boundary elements bound the solution space. An unbounded problem assumes a solution is unbounded with the electric potential φ(∞) = 0. Unbounded is true (toggle on) if the problem is an unbounded problem, false (toggle off) if the problem is bounded.

Iterations - The iterations controls the maximum number of iterations to be made by the Electric Module before exiting. The default value is 200.

Interpolation Order - The Interpolation Order controls the interpolation level on a face where the potential and/or gradient will be solved. Enter a value of 0 to specify uniform interpolation order, 1 to specify linear. An interpolation order greater than 0 provides accurate results but increases the computational expense of the calculation drastically. Only uniform (0) and linear (1) interpolation are currently available. In addition, linear interpolation is only available for 3 or 4 sided faces (i.e., triangles and quadrilaterals). If linear interpolation is specified for faces with more than four sides, the code will use uniform order interpolation. Uniform interpolation introduces only one node/face the at face center. Linear interpolation introduces n nodes for an n-sided face. Values less than zero are changed to zero and values greater than 1 are changed to 1. The default value is 0.

Expansion Order - Expansion Order controls the method of approximating the kernel function as a series expansion. Currently only Taylor series expansion of the kernel function is available. Multipole expansion is not implemented due to its lack of generality. In the future, other expansion modes may become available. The default and only possible value is 2, any changes made here will not affect the solution.

Distribution Criteria - The Dist. Criteria is the multipole accuracy parameter and controls the level of fast used in the Electric Module. A zero distribution criteria value is conventional BEM (with no attempts made to speed up the algorithm.) The solution


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will have a high level of accuracy but the solution speed will be very slow. As the parameter is increased from zero, the level of optimization increases as attempts are made to speed the level of convergence. Mechanisms, such as clumping together boundary elements into an equivalent boundary element, are used. The optimization (larger Distribution Criteria values) provides faster solutions with a lower level of accuracy of the solutions and an increase in the memory requirements to achieve a solution. Values greater than one or less than zero are changed to 0.5. The default value of 0.5 is recommended.

Relative Permittivity (εr) - Relative Permittivity is available when the unbounded option is selected. The default value is 1.0.

Capacitance Matrix

If you select the Capacitance Matrix option, CFD-ACE+ will calculate the capacitance matrix composed of multi-solid conductors and output the text file DTF_model_name.cap. A pull-down menu appears enabling you to choose the Boundary Trace Method or Volume Trace Method.

The following is a short technical note on the calculation.

Theory

Capacitors can be used to store electric potential energy. The simplest capacitor is composed of two metal plates, or conductors, facing each other. When the capacitor is charged, each plate carries an equal but opposite charge of magnitude q. The capacitance, C, is defined as:

Single Capacitor

(10-30)

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Where, V is the electric potential across the plates. The SI unit of capacitance is farad (F):

(10-31)

In a lot of cases, the farad is a very large unit. It is common to use smaller units such as microfarad (μF) or picofarad (pF):

(10-32)

(10-33)

For a system composed of several conductors, the capacitance becomes a capacitance matrix.

Capacitance Matrix

It will often write out a symmetric matrix of a capacitor system:

(10-34)

For any mutual component Cij we have:

(10-35)

and self component Cii we have:

(10-36)

Where N is the total number of conductors.


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Implementation

The numerical method to calculate a capacitance matrix has been implemented within CFD-ACE+. The actual calculation is simple. The potential on conductors will be set to 1 Volt for one conductor and 0 Volts for all other conductors at current time. CFD-ACE+ will solve the entire electric field at once and change 1 Volt potential to the next conductor and 0 Volts for all other conductors (including the conductor that has 1 Volt potential on it for the last time). This process carries to all conductors. The following chart describes the process: (1) set potential value (0, 0, 0, ..., 1, 0, 0, 0, ...0) (2) calculate electric field by solving electrostatic potential (3) calculate charge qj for each conductor (4) calculate mutual Cij (5) calculate self

(6) loop all conductors with different potential setting

Usage

Select a calculating capacitance matrix 1. Under the Electric module, select the Electrostatic option.

2. Click the Capacitance Matrix Calculation button.

3. Run CFD-ACE-Solver. The solver will create a results file named "name.cap”. Where "name” is DTF file’s name. The output format is:

Cap (1, 1) = value Conductor i1 Conductor j1 S Cap (1, 5) = value Conductor i1 Conductor j5 M Cap (i, j) = value Conductor i Conductor j M

Where, i1, j1, i, j etc. are the index of conductors, S is the self capacitance and M is the mutual capacitance.

DTF Utility Library

If the mesh is generated through the CFD-Micromesh, then prior to using the capacitance matrix calculation option, use the interconnect_tracer tool to search and link all conductors that are in focus. The tool will find those conductors that link together and create a new group of cells. Refer to interconnect_tracer below for details on how to use it. After performing this preprocess, launch CFD-GUI to set up the matrix calculation option, and run the case to obtain the capacitance matrix.

Interconnect Tracer

Function interconnect_tracer

Description

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This function uses a recursive algorithm to construct groups of contiguously connected volumes (cells) of conductor materials. New boundary and interface condition data is created from these new groups of cells and may be used to apply boundary conditions.

It is available from the DTFOL_ACE+ library as a function call and as a stand-alone application. It takes two arguments: interconnect_tracer file1.DTF file2.DTF

This will take file1.DTF, process it and create a new file, file2.DTF, which has the new volume conditions/cell groups and boundary and interface patches. It requires file1.DTF be set up to run ACE+ in a particular mode (emag?).

New interconnect_tracerT

Correct usage is: interconnect_tracerT -dtf_file file.DTF ["Seed_Patch_Name_0" "Seed_Patch_Name_1" ...] -default_name "Name"

Where:

file.DTF = Name of DTF file to modify

Seed_Patch_Name_0 = name of first patch to seed (string)

Seed_Patch_Name_1 = name of first patch to seed (string)

... = Enter any number of seed patch names.

Name = Name to be applied to all other patches created from perfect conductors (required). Example

> interconnect_tracerT -dtf_file serp3.DTF "Metal_Lin"

-default_name "Ground"

Peltier/Seebeck Effect

The Seebeck effect generates electrical potential difference in any conducting material with a temperature gradient. The value of this potential is proportional to the temperature dependent material Seebeck coefficient and temperature difference. This effect is not connected with a material junction or Joule heating effect.

When the electrical current flows through a material with a non-zero Seebeck coefficient, two additional effects are observed:

• The reversible change in heat content at an interface between dissimilar conductors, called the Peltier effect, and

• Change of heat content within the conductor, called the Thomson effect. The Seebeck coefficient is defined as [V/K]. See Theory-Peltier/Seebeck Effect for details.

Use the Peltier/Seebeck Effect 1. Activate the modeling of thermoelectric effects by selecting a solid element from the Explorer list.

2. Select MO->Electr Tab->Electric Field Options-Electrostatics option. The Peltier/Seebeck Effect option appears below the Capacitance Matrix option.

3. Select the Peltier/Seebeck option.

4. Proceed to the VC panel to define the volume conditions for the problem.


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FLOW

If the flow module has been activated in conjunction with the electric module, Lorentz Forces can be included. (The interaction of a conductor with the electric/magnetic field, J x B). To activate Lorentz Forces, click on the Lorentz Force box located in the Flow tab.

When the Electric Module is used with the Flow Module, and the Magnetic Module is not used, you must specify a constant magnetic field for the Lorentz force.

If the Magnetic Module is activated, the computed magnetic filed is used automatically by CFD-ACE+.

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Constant Magnetic Field Specification

HEAT

If you use the Heat Module with the Electric Module, the effects of Joule heating will be simulated for DC conduction, AC conduction, or transient electric model options.


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VOLUME CONDITIONS

Click the Volume Conditions [VC] tab to see the Volume Condition Panel. See Control Panel-Volume Conditions for details. Before volume conditions can be assigned, you must select an entity from the Viewer Window or the VC Explorer. Before the Piezo tab will appear, you must select the Solid option from the VC Setting Mode Properties menu.

When you select Properties from the Setting Mode pull down menu, the following properties appear: Flow, Porous Media, Momentum Resistance, Heat, Species, General, and Stress. The panel options change with each selection.

The Volume Conditions panel contains the following tabs:

Phys Struct E/M (Electric/Magnetic) Piezo (Piezoelectric)

PHYS

The Volume Conditions Phys tab contains the following options:

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• Constant - enables you to set the density as a constant value in the Rho field. • User Sub(udens) - enables you to create a user subroutine to determine the density. See User

Subroutines-Implementation-User Defined Properties or User Subroutines-User Access Routines-General Purpose Routines for details.

STRUCT The Volume Conditions Struct tab enables you to set the following options:


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• Material Type: Linear • Material: Isotropic or Anisotropic • Local X-Axis Vectors: Constant with a11 and a12 values

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• Young's Modulus: Constant with E1, E2, and E3 values • Shear Modulus: Constant with G12, G13, and G23 values • Poisson's Ratio: Constant with nu12, nu13, and nu23 values • Thermal Expansion Coefficient: Constant with Alpha1, Alpha2, and Alpha3 values

ELECTRIC/MAGNETIC

The Volume Conditions E/M tab enables you to set the following options:

Electrical Conductivity and Electrical Conductivity (Resistivity) Relative Permittivity Relative Permeability Space Charge

Electric Conductivity

The E/M tab Electric Conductivity section enables you to select an evaluation method.


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• Perfect Conductor - This method is only available when you choose Electrostatics in the MO panel (see Model Options-Electric Tab-Electric Field Options for details) and Solid Properties in the VC panel (see Volume Conditions for details). This option tells the CFD-ACE-Solver that this domain Electric Field does not exist and this domain will not be included in the solution to Poisson’s equation. This is true for both FVM and BEM problems.

• Perfect Insulator - This method is available for Electrostatics only. This method tells the CFD-ACE-Solver that this domain is a part of Poisson’s equation solution domain and electric field does exist in this domain

• Constant - When this method is selected, the constant value of electrical conductivity must be specified. • Constant Resistivity - When this method is selected, the constant resistivity of material must be specified.

CFD-ACE-Solver calculates the electrical conductivity of material as inverse of resistivity. • Function of Concentration - This option is for fluids and only available when you choose Chemistry in the

PT and MO panel, Buffer Conductivity, and the Media as Liquid Phase. (Select an entity from the Explorer List. In the MO panel, change the Chemistry Media to Liquid Phase. In the VC Panel/VC Setting Mode, in the Properties pull-down menu, change the Properties to Fluid. Then click the E/M tab and from the Electrical Conductivity (Resistivity) menu, select Function of Concentration). This option should be used for Electro-Chemistry only.

• Resistivity Function of T - When this method is selected, the reference temperature, Coefficient, and resistivity at reference temperature must be specified. The Electric Module, calculates resistivity based on local temperature using the following relation.

(10-37)

The above option is available if the Heat Transfer Module is activated.

• Resistivity Poly T - When this method is selected, specify polynomial coefficients. The Electric Module calculates resistivity based on following relation.

(10-38)

The above option is available only if you have activated the Heat Transfer Module is activated. • User Sub (uecond) - When you select this method, the solver expects a user subroutine that you have

written. For details on how to write and compile user subroutines, see User Subroutines.

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Relative Permittivity

The E/M tab Electric Conductivity section enables you to select the relative permittivity.

Only the relative permittivity (εr) of the total electric permittivity (εrεo) is set in CFD-ACE+. The value used for the permittivity of free space is εo = 8.854187 x10-12 F/m or C2/Nm2. The relative permittivity defaults to 1.0 for all cells in the grid. In the FVM, the cells are blocked so that the electric potential and field are not solved for those cells. It is important than to set boundary conditions on the faces bounding the group of cells. If fixed potential, or surface charge mixed boundary conditions are set on a face, one of the two domains the face separates must be an empty domain. The Relative Permittivity options are:

• Constant - If you select this option, enter a permeability factor in the Relative Permittivity field. • User Sub(upermittivity) - If you select the User Sub (upermittivity) option, the solver expects a user

subroutine upermittivity that you have written. For details on how to write and compile user subroutines, see User Subroutines.

Relative Permeability

The E/M tab Electric Conductivity section enables you to select the relative permeability. Only the relative permeability of the total electric permeability is set in CFD-ACE+.

• Constant- If you select this option, enter a permeability factor in the Relative Permeability field. • User Sub(urpermeability) - If you select the User Sub (urpermeability) option, the solver expects a user

subroutine urpermeability that you have written. For details on how to write and compile user subroutines, see User Subroutines.

Space Charge

The Electric/Magnetic Space Charge enables you to select the space charge evaluation method.

• Density - The space charge (ρ) is specified as a density in units of C/m3 (i.e., per unit volume). The space charge defaults to 0.0 for all cells in the grid.


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• Total - The space charge (ρ) is specified as a total charge in units of C (i.e., for whole selected volume). The space charge defaults to 0.0 for all cells in the grid.

PIEZOELECTRIC

The Piezoelectric option is available only if you have chosen MO Panel->Electric Tab->Electric Field Options->Piezoelectric. Choose this option only for those solid volumes where piezoelectric equations are to be solved. Choosing this option displays one of the following panels (2D or 3D), depending on the dimensions of the problem. Steps for using the panel are described below.

Volume Conditions - Piezoelectric Tab for 2D Model

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Volume Conditions - Piezoelectric Tab for 3D Model

Constitutive Forms

From the Constitutive Forms pull-down menu, choose the stress-charge or a strain-charge option:

Stress-charge - with this option, structural stress and the electric flux density are expressed in terms of the strain and the electric field vector as:

(10-39)

(10-40)

Strain-charge - with this option, strain and electric flux density are expressed in terms of the stress and electric field:


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(10-41)

(10-42)

where:

{σ} = Stress vector = {σ11 σ22 σ33 σ23 σ13 σ12}T(N/m2) {γ} = Strain vector = {γ11 γ22 γ33 γ23 γ13 γ12}T {E} = Electric field vector = {E1, E2, E3}T (N/C) {D} = Electric flux density vector = {D, D, D}T (C/m2)} [S] = Stiffness matrix (N/m2)

[aσ] = Stress charge piezoelectric coupling matrix (C/m2) [εσ] = Stress charge dielectric matrix (F/m) [C] = Compliance matrix (m2/N)

[aγ] = Strain charge piezoelectric coupling matrix (C/N) [εγ] = Strain charge dielectric matrix (F/m)

These equations represent the standard structural and electric field constitutive relations, with the addition of the coupling terms in which an electric field {E} produces a stress {σ} (or a strain {γ}), and a strain field {γ} (or a stress field {σ}) produces an electric flux {D}.

Conversion Between Stress Charge and Strain Charge

The stiffness matrix is obtained from information provided in the Struct section of the volume conditions and thus need not be given here. In this section, for either formulation, you must enter the dielectric matrix and the piezoelectric coupling matrix.

Inside the code, the calculations are done using the stress-charge formulation. The strain-charge formulation is input, the stress-charge data is obtained from the following transformation.

(10-43)

(10-44)

Piezoelectric materials are typically anisotropic, in which material properties depend on direction. For anisotropic materials, a local material coordinate direction for each volume is specified under the Struct property tab. The dielectric matrix and piezoelectric coupling matrix are expressed in terms of that local coordinate system. If the material was specified as isotropic, the local system is taken as the global system.

Dielectric Matrix

The dielectric matrix is a diagonal matrix and only the diagonal terms need to be given. It is expressed in terms of the local coordinate system. If the material was specified as isotropic, the local system is taken as the global system.

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Piezoelectric Coupling Matrix

The piezoelectric coupling matrix is a 6 x 3 matrix in 3D, and is 4 x 2 in 2D. The entries in this matrix must correspond to the stress (or strain) vector and electric field vector as

(10-45)

(10-46)

for 3D,

(10-47)

(10-48)

and for 2D.

The input box for the piezoelectric coupling matrix contains fields for each entry of the matrix. The subscripts 1 to 6 refer to the stress (or strain) components σ11 through σ12 as defined in Equation 10-45, and the coordinate axes X, Y, and Z correspond to the electric field directions 1, 2, and 3, respectively (Equation 10-46).

As an example, consider Equation 10-39, in which the stress vector is written in terms of the strain and the electric field. Expanding the stress contribution from the electric field (the second term on the RHS) gives:

(10-49)

(10-50)

(10-51)

(10-52)

(10-53)

(10-54)

The piezoelectric coupling matrix is typically sparse, reflecting the physical behavior of the material. As an example, a typical strain-charge piezoelectric coupling matrix for quartz is:


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From this matrix, it is seen that the electric field component E1 produces strains γ11, γ22, and γ23, and component E2 produces strains γ13 and γ12. The E3 component of the electric field vector does not contribute to the strain field.

The input for the piezoelectric coupling matrix for quartz is shown in the following figure:

Piezoelectric Coupling Matrix for Quartz

Another common piezoelectric material is PZT-5A (a lead titanate zirconate ceramic). The strain-charge coupling matrix for that material is:

For this material, the E3 electric field component produces strains γ 11, γ22, and γ33, component E2 produces strain γ23, and E1 produces γ13.

Notes

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• These properties are defined in a local material coordinate system, which must be input relative to the global system in the VC > Struct panel when the anisotropic material option is chosen. See the Volume Conditions section of the Stress Module for a description of the anisotropic material axes.

• Some references define the piezoelectric coupling matrix as the transpose of the definition used here. In that case, the matrix size from the reference would be 3x6 (for 3D) or 2x4 (for 2D). When specifying the matrix values in the GUI, use the transpose of the referenced values.

BOUNDARY CONDITIONS


The Electric Module is fully supported by the Cyclic and Arbitrary Interface boundary conditions. (See Cyclic Boundary Conditions or Arbitrary Interface Boundary Conditions for details on these types of boundary conditions and instructions for how to implement them). The Electric Module does not support the Thin Wall boundary condition feature.

All of the general boundary conditions for the Electric Module are located under the Electric (Electr) tab and can be reached when the boundary condition setting mode is set to General. The boundary conditions available in the Electric Module depend upon which Electric Field option is selected in the MO page: Ignore, Surface Charge, Fixed Potential, Fixed Current, and Dielectric. These boundary conditions are available on all computational boundaries, regardless of type (Inlet, Outlet, Wall, etc.).

The Electric Boundary Conditions include:

Ignore Fixed Potential Surface Charge Fixed Current Dielectric Piezoelectric

IGNORE

The Ignore boundary condition indicates that a set of faces has no electric boundary conditions and is ignored by the Electric Module. This boundary condition is useful when multi-disciplinary problems are being specified. Use the Ignore boundary condition to specify no electric boundary condition for a set of faces that may have a boundary condition for another problem type. This boundary condition does not require any constants to be specified. This option is available for Interfaces and for BEM on all boundary types.

FIXED POTENTIAL

The Potential (Voltage) boundary condition specifies the electric potential at a cell faces. When you select this option, enter the desired voltage (in volts) for the specified cell faces.

Voltage can be specified as constant, sinusoidal, UserSub(ubound), or parametric. See User Subroutines and Control Panel-Boundary Conditions-Parametric for more information on using the UserSub and Parametric options.


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When using Parametric settings for a CCP electrode the boundary should not be geometrically made of several segments. Parametric boundary conditions operating under identical electric boundary conditions must be made of one continuous edge (2D model) or surface (3D models).

The Sinusoid boundary condition applies time varying voltage to the surface in the following form:

When you select this option, enter values for the voltages V1 and V0 in volts, the frequency f in Hz, and φ the phase shift in degrees.

SURFACE CHARGE

The weighted (by the relative permittivity) electric field normal boundary condition specifies an electric field normal to a cell faces weighted by the absolute permittivity. When you select this option, the Surface Charge Density (ρs) field appears. Enter the charge density in V/m.

(10-55)

The default setting is ρs = 0 which is equivalent to zero normal electric field.

Surface charge can also be specified as UserSub(ubound) parametric. See User Subroutines and Control Panel-Boundary Conditions-Parametric for more information on using the UserSub and Parametric options.

FIXED CURRENT

You can specify current charge as UserSub(ubound) parametric. See User Subroutines and Control Panel-Boundary Conditions-Parametric for more information on using the UserSub and Parametric options.

This option is available only for DC Conduction Model option of the Electric Module. The current density must be specified in units of Amp/m2. This is a normal to the boundary face.

(10-56)

DIELECTRIC

When the Dielectric option is selected, faces where the voltage value is desired but the potential or its gradient is not known are included in the calculation. CFD-ACE+ automatically computes a surface charge for these boundaries. This option is valid for only CCP Plasma or the BEM solver (only at interfaces).

A dielectric boundary condition is much like a conjugate interface in heat transfer. It is simply a flux balance at the interface. The equation would be:

(10-57)

where

= permittivity of free space

= relative permittivity of cell 1 (east side cell)

= relative permittivity of cell 2 (west side cell)

= electric potential for cell 1

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= electric potential for cell 2

The normal would be the normal to the conjugate interface.

PIEZOELECTRIC

The only boundary conditions supported for piezoelectric analysis are zero gradient and specified (fixed) potential. The fixed potential value may be specified as constant, sinusoidal, parametric, or through a user subroutine.

COUPLING TO OTHER MODULES

The Electric Module is coupled to the Heat Transfer, Magnetic, Stress, and Free Surface (VOF) Modules. For the Stress and VOF Modules the coupling is automatic. To couple the influence of the magnetic field on the electric field, you must activate the src current (conduction) option under the MO/Magnet tab.

For the Stress Module, coupling is automatic when using Stress or an Implicit Pressure boundary condition with the electrostatic option of the Electric Module. If an electrostatic force exists at the boundary, it is automatically fed to the Stress Module.

Joule heating is automatically included when the electric module is coupled with the Heat Transfer Module and the DC conduction, ac conduction, or transient option is specified. See Model Options.

INITIAL CONDITIONS


The Initial Conditions can be specified as constant values or read from a previously run solution file. If constant values are specified, you must provide initial values required by the Electric Module. The values can be found under the Electric (Electr) tab and the only the initial electric potential needs to be specified.

SOLVER CONTROL

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.

The Solver Control section includes: Iter Spatial Solvers Relax


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Limits Adv

ITER

The Solver Control Iterations tab enables you to set Maximum Iterations, Convergence Criteria, and Minimum Residual. For details on these fields, see User Manual-Control Panel-Solver Controls-Iterations.

Solver Control - Iteration Tab

SPATIAL

The Solver Control Spatial tab enables you to view the type of spatial differencing being used by CFD-ACE+.

Solver Control - Spatial Tab

SOLVERS

The Solver Control Solvers tab enables you to select the linear equation solver to be used for each set of equations.

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Solver Control - Solvers Tab

Activating the Electric Module enables you to set the electric potential equation. The default linear equation solver is the conjugate gradient squared + preconditioning (CGS+Pre) solver with 50 sweeps and a convergence criteria of 0.0001. See Control Panel-Solver Selection for more information on the different linear equation solvers available. Also see Numerical Methods-Linear Equation Solvers for numerical details of the linear equation solvers.

For BEM problems, CFD-ACE-Solver uses GMRES solver. Residue is the desired residue reduction factor for the electric potential. The residue is normalized to one at the beginning of the calculation. The calculation stops when either the desired electric potential residue is achieved or the maximum iterations is exceeded. The default value is 1.0 x10-4.

The matrix-less Gmres method is used to solve the equation systems iteratively. A diagonal precondition is used everywhere except on the interfaces, which uses a 2x2 submatrix precondition. The Gmres Restart Steps parameter controls the number of sweeps within the linear solver for each iteration. The default value is 10.

RELAX 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 electric equations.

Solver Control - Relax Tab

Activating the Electric Module enables you to set the electric potential dependent variable. See Under Relaxation Parameters for details on the mechanics of setting the under relaxation values. See Solution Methods-Under Relaxation for numerical details of how under relaxation is applied.


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The electric potential equation uses an inertial under relaxation scheme and the default value is 0.0001. Increasing this value applies more under relaxation and therefore adds stability to the solution at the cost of slower convergence.


LIMITS The Solver Control Limits tab enables you to 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 Electric Module enables you to set limits for the electric potential variable. See Control Panel-Variable Limits for details on how limits are applied.

Solver Control - Limits Tab

ADV The Solver Control Advanced tab appears when you select the Boundary Element Method in the Model Options Electric tab.

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Advanced tab in the Solver Control Panel in Electric Module Settings (for BEM Solver) Mode

OUTPUT There are no settings under the Output tab that affect the Electric Module. See Control Panel-Output Options for details about the available output settings. The Output section includes Graphical Output and Summary 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 viewing and analysis in CFD-VIEW. Activating the Electric Module provides output of the variables listed below:

Electric Module Graphical Output Variable Units

Electrical Conductivity Ω-1m-1

Relative Permeability -

Relative Permittivity -

Electric Field N/C

Surface Charge C/m2

Electric Virtual Force Vector N/m3

Electrostatic Pressure Force N/m2


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Electric Potential J/C

Conduction Current Density A/m2

SUMMARY OUTPUT

Under the Summaries tab, select the summary information to be written to the main ASCII text output file (modelname.out). Activating the Electric Module gives the option of outputting the Electric Flux Summary for conduction problems, and writing the current summary to the main output file. See the Control Panel-Summary Output for details on the general summary output options, including boundary integral output, diagnostics output, and monitor point output.

POST PROCESSING

CFD-VIEW can post-process the solutions. When the Electric Module is invoked, the electric potential field is usually of interest. The electric potential field can be visualized with surface contours and analyzed through the use of point and line probes. complete list of post processing variables available as a result of using the Electric Module is shown in the table.

Post Processing Variables


efieldx Electric Field, x-component N/C

efieldxi Electric Field, imaginary x-component N/C

efieldy Electric Field, y-component N/C

efieldyi Electric Field, imaginary y-component N/C

efieldz Electric Field, z-component N/C

efieldzi Electric Field, imaginary z-component N/C

el_epsr Dielectric constant or relative permittivity -

el_pot Electric Potential Volt

epsr_x_En Weighted electric field normal N/C

Jcx_i Conduction current density, imaginary x-

component A/m2

Jcx_r Conduction current density, real x- A/m2

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component

Jcy_i Conduction current density, imaginary y-

component A/m2

Jcy_r Conduction current density, real y-

component A/m2

Jcz_i Conduction current density, imaginary z-

component A/m2

Jcz_r Conduction current density, real z-component A/m2

mu_r Relative Permeability -

pestat Electrostatic pressure N/m2

Qsurf Surface charge density C/m2

Qvol Volume Charge density C/m3

sig_i Imaginary part of conductivity Ω-1m-1

sig_r Real part of conductivity Ω-1m-1

vfx_estat Virtual electric force, x-component N/m3

vfy_estat Virtual electric force, y-component N/m3

vfz_estat Virtual electric force, z-component N/m3

Symbol Definitions

The following are commonly used symbols, definitions, and units. A bold symbol denotes a vector quantity.

Symbol Definition Unit

ε electric permittivity C2/Nm2

εr electric relative permittivity

(dielectric constant) -


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εo electric permittivity of free

space (8.854187 x10-12) C2/Nm2

φ electric potential J/C

electric potential phasor

(complex number) J/C

ρ space charge density C/m3

resistivity Ω m

ρs surface charge density C/m2

ρ electric conductivity Ω-1m-1

ω radian frequency (2πf) rad/s

A magnetic vector potential N/A

D displacement flux density

vector C/m2

E electric field vector N/C

J total current density vector A/m2

Jc conduction current density

vector A/m2

Q total charge C

f frequency 1/s

j imaginary number often

viewed as -

References

Bachtold, Martin. "Efficient 3D computation of Electrostatic Fields and Forces in Microsystems." Diss. Swiss Federal Institute of Technology Zurich. 1997.

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Vágó, I. and Gyimesi, M. “Electromagnetic Fields.” Akadémiai Kiadó, Budapest. 1998. pp. 60-63.


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Appendix A Post-Processing Variables by Module

Biochemistry Module Variable Description Units

D_AnalyteName Diffusivity m2/s

Surfcon_AnalyteName Surface Concentration moles/m2

AnalyteName Analyte Concentration M

Irrcon_AnalyteName Irreversible Analyte Concentration M

Reaction_Rate Generation Rate of Product/consumption

Rate of Substrate

moles/m>2/sec

Wall_conc_AnalyteName Wall concentration M

Cavitation Module Variable Description

MassFr Mass fraction

Total_Volume_Fraction Total volume fraction

Vapor_Volume_Fraction Vapor volume fraction

Chemistry Module Variable Description

Nox_Rate Nox production rate

Progress Progress Variable

React_Rate Reaction Rate

Species name Species Mass fraction

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Electric Module Variable Description Units

efieldx Electric Field, x-component N/C

efieldxi Electric Field, imaginary x-component N/C

efieldy Electric Field, y-component N/C

efieldyi Electric Field, imaginary y-component N/C



efieldzi Electric Field, imaginary z-component N/C

el_epsr Dielectric constant or relative permittivity -

el_pot Electric Potential Volt

epsr_x_En Weighted electric field normal N/C

Jcx_i Conduction current density, imaginary x-

component

A/m2

Jcx_r Conduction current density, real x-component A/m2

Jcy_i Conduction current density, imaginary y-

component

A/m2

Jcy_r Conduction current density, real y-component A/m2

Jcz_i Conduction current density, imaginary z-

component

A/m2

Jcz_r Conduction current density, real z-component A/m2

mu_r Relative Permeability -

pestat Electrostatic pressure N/m2

Qsurf Surface charge density C/m2

Qvol Volume Charge density C/m3


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sig_i Imaginary part of conductivity Ω-1m-1

sig_r Real part of conductivity Ω-1m-1

vfx_estat Virtual electric force, x-component N/m3

vfy_estat Virtual electric force, y-component N/m3

vfz_estat Virtual electric force, z-component N/m3

Flow Module Variable Description Units

Mach Mach Number -



RHO Density Kg/m3

STRAIN_RATE Strain 1/s

Stream_Function Stream Function Kg/m3

U, V, W X-direction Velocity, Y-direction velocity, Z-

direction Velocity

m/s

U_absolute,

V_absolute,

W_absolute



CFL_Number CFL Number* -

WallViscousStress_X,

WallViscousStress_Y,

WallViscousStress_Z

Wall Viscous Stress N/m2

WallViscousStressMagnitude Wall Viscous Stress Magnitude N/m2

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WallShearStress_X,

WallShearStress_Y,

WallShearStress_Z

Wall Shear Stress N/m2

WallShearStressMagnitude Wall Shear Stress Magnitude N/m2

SkinFrictionCoefficient Skin Friction Coefficient -

PressureCoefficient Pressure Coefficient -

Vorticity Vorticity Criteria 1/s

Vis Effective Viscosity kg/m/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


Heat Module Variable Description Units

COND 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


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T_TOT Total Temperature K

Wall_Heat_Flux Wall heat Flux W/m2

Wall_Rad_Flux Wall radiative flux W/m2

Magnetic Module Variable Description Units

Ax_r Magnetic Vector Potential Real x Component Wb/m

Ax_i Magnetic Vector Potential Imaginary x Component Wb/m

Ay_r Magnetic Vector Potential Real y Component Wb/m

Ay_i Magnetic Vector Potential Imaginary y Component Wb/m

Az_r Magnetic Vector Potential Real z Component Wb/m

Az_i Magnetic Vector Potential Imaginary z Component Wb/m

J_eddy Eddy current A/m2

Power_dissipation Power dissipation W/m3

Ex_i RF electric field, imaginary-x component N/C

Ey_i RF electric field, imaginary-y component N/C

Ez_i RF electric field, imaginary-z component N/C

|E|rf Magnitude of rf electric field N/C

Bx_r Magnetic field real-x component N/A-M

By_r Magnetic field real-y component N/A-M

Bz_r Magnetic field real-z component N/A-M

Bx_i Magnetic field imaginary-x component N/A-M

By_i Magnetic field imaginary-y component N/A-M

Bz_i Magnetic field imaginary-z component N/A-M

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vfx_mag Virtual magnetic force, x-component N

vfy_mag Virtual magnetic force, y-component N

vfz_mag Virtual magnetic force, z-component N

fx_mag Magnetic force, x-component N

fy_mag Magnetic force, y-component N

fz_mag Magnetic force, z-component N

Plasma Module Variable Description Units

Te Electron temperature eV

Ne Electron number density 1/m3

nu Collision frequency s-1

Jx_Ne Current density, x-component A/m2

Jy_Ne Current density, y-component A/m2

Jz_Ne Current density, z-component A/m2

Ne_avg Average Electron number density m-3

Te_avg Average electron temperature eV

Pwr_avg Average power W/m3

Mob_e Electron mobility m2/(V-s)

dep_avg Deposition rate kg/m2s

Power Power W/m3

E_x x component of ambipolar field V/m

E_y y component of ambipolar field V/m

E_z z component of ambipolar field V/m


400

Scalar Module Variable Description Units

D_ScalarName Scalar Diffusion Coefficient kg/m-s

ScalarName Scalar Name -

Spray Module Variable Description Units

spr_src_u u-momentum source term kg m/s2

spr_src_v v-momentum source term kg m/s2

spr_src_w w-momentum source term kg m/s2

spr_src_h enthalpy source term J/s

spr_src_m mass source term kg/s

spr_volfrac spray volume fraction -

spr_dist particle concentration particles/m3

Stress Module Variable Description Units

TauMax Maximum shears stress N/m2

Sigmazz Cartesian stress component σzz N/m2

VonMises VonMises Stress N/m2

Sigmaxx Cartesian stress component σxx N/m2

Sigmaxy Cartesian stress component σxy N/m2

Sigmaxz Cartesian stress component σxz N/m2

strainxx Cartesian strain component εxx -

strainyy Cartesian strain component εyy -

strainzz Cartesian strain component εzz -

Modules

401

strainxy Cartesian strain component εxy -

strainxz Cartesian strain component εxz -

strainyz Cartesian strain component εyz -

EpsMax Maximum principal strain -

EpsMin 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

Turbulance Module Variable Description Units

D Dissipation Rate(κ-ε model)

Specific rate of dissipation (κ-ε model)

m2/s3

s-1

K Kinetic energy m2/s2

VIS_T Turbulent Viscosity kg/m-s

YPLUS Yplus values -

Rg Grid Parameter -

K_length Kolmogorov length scale m

K_time Kolmogorov time scale s

K_velocity Kolmogorov velocity scale m/s

Two-Fluid Module Variable Description Units


402

U2 X-direction velocity of 2nd fluid m/s

V2 Y-direction velocity of 2nd fluid m/s

W2 Z-direction velocity of 2nd fluid m/s

RHO2 Density of 2nd fluid kg/m3

H2 Enthalpy of 2nd fluid kg-m2/s2

Visc2 Viscosity of 2nd fluid kg/m-s

T2 Temperature of 2nd fluid K

Alpha Volume fraction of 2nd fluid -

Stream2 Stream Function of 2nd fluid -

VOF Module Variable Description Units

LiqVOF Volume fraction of 2nd fluid -

VOFOld Volume fraction at previous time step -

Rho_1 Density of fluid 1 kg/m3

Rho_2 Density of 2nd fluid kg/m3

curvtur curvature m-1

Modules

403


404

Appendix B Post-processing Engineering Quantities by Module

FLOW

Pressure Coefficient Description Surface local pressure coefficient

Post-processing variable name PressureCoefficient

Unit None

Equation

Wall Viscous Stress Description Surface local wall viscous stress

Post-processing variable name WallViscousStress_X/Y/Z and WallViscousStressMagnitude

Unit N/m2

Equation

where:

τviscous is the local wall viscous stress vector I is the identity tensor n is the local outward-pointing surface normal unit vector

Wall Shear Stress

Description Surface local wall viscous shear stress, i.e. component of wall

viscous stress vector tangential to the surface.

Post-processing variable names WallShearStress_X/Y/Z and WallShearStressMagnitude

Unit N/m2

Equation

Modules

405

where:

τshear is the local wall viscous shear stress vector n is the local outward-pointing surface normal unit vector

Skin Friction Coefficient Description Surface local skin friction coefficient

Post-processing variable names SkinFrictionCoefficient

Unit None

Equation

where:

ρ is the local fluid density

HEAT

Heat Transfer Coefficient Description Surface local heat transfer coefficient

Post-processing variable names HeatTransferCoefficient

Unit W/(m2K)

Equation

where:

qw is the local wall heat flux Tw is the local wall temperature T is either a fixed user-defined value (Tref), or the local fluid temperature (i.e. the temperature on the first cell at the wall)

Nusselt Number Description Surface local Nusselt number

Post-processing variable names NusseltNumber

Unit None


406

Equation

Prandtl Number Description Volume local Prandtl number

Post-processing variable names PrandtlNumber

Unit None

Equation

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