# Diffusion term

## Discretisation of the Diffusion Term

### Description

For a general control volume (orthogonal, non-orthogonal), the discretization of the diffusion term can be written in the following form
$\int_{S}\Gamma\nabla\phi\cdot{\rm{d\vec S}} = \sum_{faces}\Gamma _f \nabla \phi _f \cdot{\rm{\vec S_f}}$
where

• S denotes the surface area of the control volume
• $S_f$ denotes the area of a face for the control volume

As usual, the subscript f refers to a given face. The figure below describes the terminology used in the framework of a general non-orthogonal control volume

A general non-orthogonal control volume

Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes.
A control volume in mesh is made up of set of faces enclosing it. Where $S_f$ represents the magnitude of area of the face. And n represents the normal unit vector of the face under consideration.

If $\vec r_{P}$ and $\vec r_{N}$ are position vector of centroids of cells P and N respectively. Then, we define
$\overrightarrow{d_{PN}}= \vec r_{P} - \vec r_{N}$

We wish to approaximate the diffusive flux $D_f = \Gamma _f \nabla \phi _f \cdot{\rm{\vec S_f}}$ at the face.

### Approach 1

A first approach is to use a simple expression for estimating the gradient of a scalar normal to the face.

$D_f = \Gamma _f \nabla \phi _f \cdot \vec S_f = \Gamma _f \left[ {\left( {\phi _N - \phi _P } \right)\left| {{{\vec S_f} \over {\overrightarrow{d_{PN}}}}} \right|} \right]$

where $\Gamma _f$ is a suitable face average.

This approach is not very good when the non-orthogonality of the faces increases. If this is the case, it is advisable to use one of the following approaches.

### Approach 2

We define the vector $\vec \alpha {\rm{ = }}\frac{{{\rm{\vec {S_f}}}}}{{{\rm{\vec S_f}} \cdot {\overrightarrow{d_{PN}}}}}$

giving us the expression:

$D_f = \Gamma _f \nabla \phi _f \cdot{\rm{\vec S_f = }}\Gamma _{\rm{f}} \left[ {\left( {\phi _N - \phi _P } \right)\vec \alpha \cdot {\rm{\vec S_f + }}\bar \nabla \phi_f \cdot {\rm{\vec S_f - }}\left( {\bar \nabla \phi_f \cdot {\overrightarrow{d_{PN}}}} \right)\vec \alpha \cdot {\rm{\vec S_f}}} \right]$

where $\bar \nabla \phi _f$ and $\Gamma _f$ are suitable face averages.

## References

1. Ferziger, J.H. and Peric, M. (2001), Computational Methods for Fluid Dynamics, ISBN 3540420746, 3rd Rev. Ed., Springer-Verlag, Berlin..
2. Hrvoje, Jasak (1996), "Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows", PhD Thesis, Imperial College, University of London (download).