# Diffusion term

## Discretisation of Diffusive Term

### Description

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. The figure 1.1 shows a typical situation. Where A represent the magnitude of area of the face. And n represents the normal unit vector of the face under consideration.

Figure 1.1

$\vec r_{0}$ and $\vec r_{1}$ are position vector of centroids of cells cell 0 and cell 1 respectively.
${\rm{d\vec s}} = \vec r_{1} - \vec r_{0}$

We wish to approaximate $D_f = \Gamma _f \nabla \phi _f \bullet {\rm{\vec A}}$ at the face.

### Approach 1

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

$D_f = \Gamma _f \nabla \phi _f \bullet \vec A = \Gamma _f \left[ {\left( {\phi _1 - \phi _0 } \right)\left| {{{\vec A} \over {d\vec s}}} \right|} \right]$

where $\Gamma _f$ is suitable face averages.

This approach is not very good when the non-orthogonality of the faces increases. Instead for the fairly non-orthogonal meshes, it is advisable to use the following approaches.

### Approach 2

We define vector $\vec \alpha {\rm{ = }}\frac{{{\rm{\vec A}}}}{{{\rm{\vec A}} \bullet {\rm{d\vec s}}}}$

giving us the expression:

$D_f = \Gamma _f \nabla \phi _f \bullet {\rm{\vec A = }}\Gamma _{\rm{f}} \left[ {\left( {\phi _1 - \phi _0 } \right)\vec \alpha \bullet {\rm{\vec A + }}\bar \nabla \phi \bullet {\rm{\vec A - }}\left( {\bar \nabla \phi \bullet {\rm{d\vec s}}} \right)\vec \alpha \bullet {\rm{\vec A}}} \right]$

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

## Reference

1. Ferziger, J.H. and Peric, M. 2002. "Computational Methods for Fluid Dynamics", 3rd rev. ed., Springer-Verlag, Berlin.
2. Hrvoje Jasak, PhD. Thesis, "Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows "