### MathematicsNumericsDerivationsAndOpenFOAM.pdf

MathematicsNumericsDerivationsAndOpenFOAM.pdf

File Size:

1.69 MB

Version:

4.0

License:

Email:

Tobias[dot]Holzmann[at]holzmann-cfd[dot]de

Date:

02 May 2016

### Researchgate: https://www.researchgate.net/profile/Tobias_Holzmann/publications

Soft-cover copy can be ordered by sending an email to Tobias.Holzmann@Holzmann-cfd.de

Features

### 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 d
*t*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 eqn. 7.4, 7.5, 7.6, 7.7 and 7.8
- Corrected eqn. 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- Added chapter about the bulk viscosity
- 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
- Added new chapter about "OpenFOAM(R) tutorials"
- Updated and changed a lot of sentences for a better understanding in the whole documentation
- Added literature
- Added "how to cite"
- 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
- Added SIMPLEC to SIMPLE
- Added SIMPLEC to PIMPLE

### 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)
- Added non-symmetric tensor definition
- 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)
- Increased readability drastically
- 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
- Added one literature
- 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(R)

- 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 p*, important to know in some cases)
- 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 (1.15) dyadic sign added
- Equation (1.16) dyadic sign added
- 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 (2.30), (2.31) and (2.32) dyadic sign added
- 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)

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