# Mathematics, Numerics, Derivations and OpenFOAM®

## The Edition for the OpenFOAM® Community (for OpenFOAM® v7)

The well-known book by Holzmann CFD. This version of the book, reviewed and rewritten for OpenFOAM® version 7, is available for some fee. The prior version was read and downloaded more than 35.000 on the Holzmann CFD website and ResearchGate. The book enjoys world-wide acceptance and was released in November 2019. This edition will be updated and maintained.

- Description
- REVIEWS

Dedicated to the OpenFOAM® community and to all colleagues and people who support my work on Holzmann CFD. The ambition to write the book was based on my personal love for the open-source thought. Thus, my objective is to give any interested CFD lover an introduction to computational fluid dynamics while showing interesting equations and some relations which are not given — or at least I did not realize them — in most of the famous books and papers in that particular 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

This book gives an introduction to the underlying mathematics used in the field of computational fluid dynamics. After presenting the mathematics aspects, all conservation equations are derived using a finite volume element, dV. It is precisely described how the mass and momentum equation can be obtained. Subsequently, all kinds of the energy equations 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 afterward and it is shown how this equation can be used to derive other ones.

The subsequent 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 entirely 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 epsilon are deducted. The topic ends with a brief description of 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 are said.

Finally, a more general discussion of the different pressure-momentum coupling algorithms is presented. Subsequently, the PIMPLE-algorithm is explained while considering an OpenFOAM® case.

The last chapter is related to OpenFOAM® beginners which are seeking tutorials and some other useful information and websites.

## Abstract

This book gives an introduction to the underlying mathematics used in the field of computational fluid dynamics. After presenting the mathematics aspects, all conservation equations are derived using a finite volume element, dV. It is precisely described how the mass and momentum equation can be obtained. Subsequently, all kinds of the energy equations 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 afterward and it is shown how this equation can be used to derive other ones.

The subsequent 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 entirely 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 epsilon are deducted. The topic ends with a brief description of 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 are said.

Finally, a more general discussion of the different pressure-momentum coupling algorithms is presented. Subsequently, the PIMPLE-algorithm is explained while considering an OpenFOAM® case.

The last chapter is related to OpenFOAM® beginners which are seeking tutorials and some other useful information and websites.

### Changes from Release 5.x (04.04.2018) to Release 7.x (11.2019)

- Updated the design
- Corrected typos, wording, and the language
- Added the derivation of the temperature equation for solids
- Renewed the chapter "Calculation of the Shear-Rate Tensor in OpenFOAM®
- Renewed the chapter "The numerical algorithms: SIMPLE, PISO and PIMPLE"
- Split the book in two/three versions
- Green Free Version Limited - Includes all changes while some pages where faded out
- Green Paid Version - Includes all changes
- Blue Paid Version - All new book extensions will be available here (in Progress)