Mathematics, Numerics, Derivations and OpenFOAM®

The publication was started in 2015 and is also available on ResearchGate. Tobias tries to keep the book up-to-date.

» Release notes ...

The publication named »Mathematics, Numerics, Derivations and OpenFOAM®« is the outcome of Tobias' time during his Ph.D. at the Montanuniverität Leoben (Austria). In the beginning, the book was more like a summary of different equations. After the community got a 30-page outline, a lot of feedback about an extension was given. Thus, Tobias decided to go on writing the summary, which ended in the actual book.

The book can be translated into all languages using the official repository on GitHub

The book contains the general equations used for computational fluid dynamics. However, compared to other literature, Tobias made the derivation of the equations fundamental and showed every single step. A volume element is used to demonstrate the origin of the mass conservation, momentum equation, and the different kinds of energy equations. Additionally, the transformation of the equations from the Cartesian / Einstein-summation form to the vector and integral form is described as well as the conversion from conservative (Euler) to non-conserved (Lagrangian) is prescribed.

Several hundred hours are already spend for writing the book

Tobias tries do extend the book from time to time and keep the material and code snippets that are presented in the book up to date to the OpenFOAM Foundation version. The free version is a present to the OpenFOAM® community. Tobias hopes that people understand that the full version does cost a bit as people otherwise do not support his work.

The book is related to people who

  • are interested in the fundamental equations
  • are interested in the basic Reynolds-Averaging approach
  • are interested in the closure problem
  • want to see the derivations of equations
  • are interested in the PIMPLE algorithm
  • are interested in the divDevREff and divDevRhoREff function
  • are interested in some OpenFOAM® classes
  • are interested in how to transfer equations to the OpenFOAM® framework
  • are interested in general computational fluid dynamics knowledge
  • ... just check the free version

The book is not related to

  • fundamental OpenFOAM® programming
  • a detailed description of the usage of OpenFOAM®
  • to turbulence modeling, e.g., different turbulence models and their limitations
  • Numerical schemes and their implementation
  • A guide of best practice to use OpenFOAM®
  • advanced visualization using DAKOTA®
  • any tutorial published by Tobias on this website
  • other third party software
  • boundary condition handling
  • common practice guide for CFD analysis

The free version shows the whole content of the book except some unique parts which took a lot of time of the investigation

A single page from the book
A single page from the book
A single page from the book

Tobias provides the LaTeX files on the new GitHub account

The new vision of Tobias is to make his material freely available for everybody and forever. Thus, he decided to publish the LaTeX files for the book related to the free version. The paid one is still in a private repository. However, this allows the community to do the following:


  • Translate the book using a new branch
  • Extend different topics and support the community
  • Remove mistakes directly while making a pull request
  • Having direct access to the files and repository allows transparency in changes (commits)

Released on 04.01.2021 — Downloads:


Github URL:
Written by Tobias Holzmann
Support the work of Tobias Holzmann