### Mathematics, Numerics, Derivations and OpenFOAM®

Mathematics, Numerics, Derivations and OpenFOAM®
MathematicsNumericsDerivationsAndOpenFOAM.pdf
File Size:
1.75 MB
Version:
5.x
Author:
Email:
Tobias[dot]Holzmann[at]Holzmann-cfd[dot]de
Date:
03 November 2017

# OpenFOAM® Community

Dedicated to the OpenFOAM® community and especially to all colleagues and people who support me. The ambition to write the book is based on my personal love to the open source thought. Thus, my objective is to give you an introduction to computational fluid dynamics, show interesting equations and some relations which are not given in most of the books and papers which are famous in that area. In addition, the book should prepare you for the tasks that you may work on during your personal career, hopefully with OpenFOAM®.

DOI: 10.13140/RG.2.2.27193.36960

## Abstract

This book gives an introduction to the basic mathematics used in the field of computational fluid dynamics. After presenting the mathematic aspects, all conservation equations are derived using a finite volume element, dV . It is precisely described how the mass and momentum equation can be derived. Subsequently, all kinds of energy equation are discussed and presented namely the kinetic energy, internal energy, total energy and the enthalpy equation. Based on the nature of the equations, the general governing equation is introduced afterwards and it is shown how this equation can be used to derive other ones. The following chapters discuss the definition of the shear-rate tensor τ for Newtonian fluids and is followed by a discussion that shows the analogy between the Cauchy stress tensor σ, the shear-rate tensor τ and the pressure p. All equations are summed up with a one page summary at the end. Based on the fact that engineering applications are mostly turbulent, the Reynolds-Averaging methods are presented and explained. Subsequently the incompressible equations are derived and finally the closure problem is discussed in detail. Here, the Reynolds-Stress equation — which is fully derived in the appendix — and the analogy to the Cauchy stress tensor is shown. To close the subject of turbulent flows, the eddy-viscosity theory is introduced and the equation for the turbulent kinetic energy k and dissipation  are deducted. The topic ends with a brief description about the derivation for the compressible Navier-Stokes-Equations equations and its difficulties and validity. The last chapters of the book are related to the detailed explanation of the implementation of the shear-rate tensor calculation in OpenFOAM® . During the investigation into the C++ code, the mathematical equations are given and a few words about the numerical stabilization is said. Finally, a more general discussion of the different pressure-momentum coupling algorithms is given. Subsequently, the PIMPLE-algorithm is explained while considering an OpenFOAM® case. The last chapter is related to OpenFOAM® beginners which are seeking for tutorials and some other useful information and websites.

Changelog

### Changes from edition 4 (24.09.2017) to - Release 5.x (04.04.2018)

• Updated the c++ code to the OpenFOAM-5.x Foundation version (not yet done)
• Version number of the book correlates to the Foundation version (not yet)
• Correction of nOuterCorrectors set-up on page 115. I mentioned to set this quantity to zero to get the PISO loop but it is one
• On page 116 the code tells one that the turbOnFinalIter is set to true but I mentioned that it is set to false in the text
Thanks to Hossein Heydari for the report

• Summary of equations. The enthalpy equation was wrong
• Added other equations to the summary
Thanks to Dr. Lars Korn for the hints

• First term in equation 2.61 corrected (density was missing)
Thanks to Chris Harding for the hint

• Corrected typos and sentences
• Improved English
• Book format changed
• Bookmarks for PDF corrected

### Changes from edition 4 (20.09.2017) to edition 4 (24.09.2017)

• Sign mistake in appendix page 155 Shear-Rate Term a)
• Sign mistake in appendix page 159 equation 14.32 f)
• Derivative wrong in equation 14.42, 14.43 and 14.44 (second term on the LHS)
• Minor grammar mistakes
Thanks to Vishwesh Ravi Shrimali for reporting the mistakes

### Changes from edition 4 (17.09.2017) to edition 4 (20.09.2017)

• Equation 9.97, 9.98, 9.99 9.100, 9.101 average symbol missing on some quantities
• Equation 9.107 and 9.108 had a wrong derivation
• Equation 9.110 had a wrong diffusion term
• Equation 14.18 and 14.19 denominator was wrong
Thanks to Vishwesh Ravi Shrimali for reporting the mistakes

### Changes from edition 4 (03.09.2017) to edition 4 (17.09.2017)

• Essential error in integral form of the momentum equation 2.28 and 8.8: The pressure term is a surface integral
• Typos correction on page 23 and 49
Thanks to Florian Kunstmann for the hints
• Missing absolute bar in page 50 (summary in the total energy equation
• Equation 5.26, the term on the left hand side was wrong (the shear-rate term)
• Equation 9.39, 9.41, 9.46, 9.47 and 9.48 were wrong (pressure term with wrong sign)
• Equation 9.7 missing dt term on the LHS
• Equation 9.51 and 9.52  represents one equation (not two)
• Equation 9.64 (new number 9.63), missing identity on the RHS (underbrace)
• Equation 9.73 (new number 9.72), missing trace operator in second line
• Equation 9.83 (new number 9.82), pressure typo (star sign has to be applied)
Thanks to Vishwesh Ravi Shrimali for the hints (highly appreciated)
• Equation 9.78, 9.79, 9.80 (new number 9.77, 9.78 and 9.79), missing bar on pressure

Note: Based on merging 9.51 and 9.52 all following equations are reduced by the number of one.

### Changes from edition 4 (29.07.2017) to edition 4 (03.09.2017)

• Corrected equation 2.95 (density inside the squares)
Thanks to Vishwesh Ravi Shrimali for the hint

### Changes from edition 4 (24.07.2017) to edition 4 (29.07.2017)

• Page 69, section 9.1the sentence including  "... figure 7.1" was corrected (number of the figure was wrong)
• Page 70, equation 9.3 corrected
• Page 86, equation 9.64 corrected
Thanks to Vigneshwaran Sankar for the hints
• Page 115, PISO is done with nOuterCorrectors equal one (previous ... equal zero)
Thanks to Arnav Ajmani for the hint
• New chapter about the relaxation techniques in OpenFOAM® (field and matrix) to get more information about the information loose and the Final flag

### Changes from edition 4 (21.02.2017) to edition 4 (24.07.2017)

• Changed the word viscose to viscous (thanks to my colleague Christian Rodrigues)

### Changes from edition 4 (29.01.2017) to edition 4 (21.02.2017)

• Changes only related to chapter 7
• Improved English
• Added dependency to equation 7.4, 7.5, 7.6, 7.7 and 7.8
• Corrected equation 7.16, 7.17 and 7.18

### Changes from edition 4 (26.11.2016) to edition 4 (29.01.2017)

• Some typos
• residualControls changed to residualControl in the chapter of PIMPLE
• Big changes in chapter 4 and 5 according to the bulk viscosity
• Some sentence correction in chapter 10
• Corrected equation 10.12
• Added chapter 12 - OpenFOAM® tutorials

### Changes from version 4 (October 19.10.2016) to edition 4 (26.11.2016)

• Corrected a lot of typos
• Renewed the introduction of the mass conservation equation
• Renewed the introduction of the momentum conservation equation
• Corrected the summary page (equations were wrong)
• Corrected equation 2.46
• Updated and changed a lot of sentences for a better understanding in the whole documentation
• Renewed the abstract
• Book read (for me) to print

### Changes from version 4 (October 04.10.2016) to Revision 4 (October 19.10.2016)

• Added subsection for the usage of the SIMPLEC algorithm

### Changes from version 4 (September) to Revision 4 (October)

• Implemented English correction in chapter 1 (thanks to Vignesh T.G.)
• Corrected Gauss summation convention (thanks to Jaap Versteegh)
• Extended equation (1.13) to clear some doubts
• Included hyperref to the document
• Re-format the code listings
• New literature style
• Added a new chapter (23 pages) about the usage of the PIMPLE algorithm (community wish)
• Some typo correction

### Changes from version 3 to Revision 4 (September)

• Corrected figure 2.2
• Corrected two terms (plus sign of pressure to minus sign)
• Moved Reynolds-Stress equation derivation to the appendix
• Summary of equation is now a own chapter
• Changed format to B5 (for book)
• Generated a book cover
• All formula's are included in the text flow (they end with a point or comma)
• A lot of equations are new formated

### Changes from version 2.1 to 3.0

• Extended ~ 35 pages for turbulence modeling
• Correct some mistakes (in general)
• Include the handling of the time derivative as differences first
• English improvements
• Extend and reorganized the mathematic chapter
• Reorganized the chapters
• Reorganize the shear-rate tensor derivation in OpenFOAM®
• Turbulence modeling
• Introduce to Reynolds-Averaging methods (time-average, ensemble, ...)
• Time-Averaging operations on linear, nonlinear terms and constants
• Derivation of incompressible Reynolds time-averaged mass conservation
• Derivation of compressible Reynolds time-averaged mass conservation (to show why we get more troubles here)
• Introduce the turbulent Reynolds stress tensor
• Derivation of incompressible Reynolds time-averaged momentum equation
• Introducing the closure problem
• Comparison of Reynolds stress tensor and Cauchy stress tensor (especially to show the modified pressure we use in OpenFOAM®)
• Introducing to the Boussinesq eddy viscosity approximation
• Introducing the eddy viscosity approximation
• Derivation of the incompressible Reynolds-Stress equation to derive the turbulent kinetic energy equation
• Derivation of the incompressible kinetic energy equation of the turbulence
• Relation between length scale and dissipation
• Derivation of the incompressible dissipation rate equation (not shown in detail)
• Introducing the Favre averaging concept
• Demonstrate the concept using the compressible mass conservation equation

### Changes from version 2.0 to 2.1

• English improvements and corrections corrected (thanks to Sergei for the nice corrections)
• Gauss theorem introduced in the mathematic section
• Equation (1.6) corrected
• After equation (1.7), the vector notation (brackets) was wrong
• Equation (1.14) results not in a vector, it results in a scalar
• Equation (2.4) corrected again
• The sentence after (2.9) was changed to be more clear
• Section 2.1.1, remarks and addition information to gauss theorem etc.
• Equation (2.13) - (2.16) corrected; velocity vector was not bold
• Literature added for the non-common continuity equation
• Equation (2.77) other re-arrangement, to be more clear what we do
• Equation (2.78) conversation -> conversion
• Equation (2.85) 4th term on the RHS is a surface integral (changed integral sign)
• Added a description why we use the small bullet in the integrals instead of the big bullet (because both represent the inner product of two vectors)

Version 3, 29 June 2007

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If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. “Knowingly relying” means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

A patent license is “discriminatory” if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

#### 12. No Surrender of Others' Freedom.

If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

#### 13. Use with the GNU Affero General Public License.

Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

#### 14. Revised Versions of this License.

The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License “or any later version” applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

#### 15. Disclaimer of Warranty.

THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

#### 16. Limitation of Liability.

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

#### 17. Interpretation of Sections 15 and 16.

If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

END OF TERMS AND CONDITIONS

### How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

```    <one line to give the program's name and a brief idea of what it does.>
Copyright (C) <year>  <name of author>

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <https://www.gnu.org/licenses/>.
```

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

```    <program>  Copyright (C) <year>  <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
```

The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see <https://www.gnu.org/licenses/>.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read <https://www.gnu.org/licenses/why-not-lgpl.html>.

I agree to the terms listed above

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