# Diffusion term

### From CFD-Wiki

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==Discretisation of Diffusive Term == | ==Discretisation of Diffusive Term == | ||

- | === | + | === Description=== |

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- | + | Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes. | |

- | + | <br> | |

- | + | ||

- | + | ||

- | + | ||

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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. | 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. | ||

<br> | <br> | ||

[[Image:Nm_descretisation_diffusionterms_01.jpg]] <br> | [[Image:Nm_descretisation_diffusionterms_01.jpg]] <br> | ||

'''Figure 1.1''' <br> | '''Figure 1.1''' <br> | ||

+ | :<br> | ||

<math> \vec r_{0} </math> and <math> \vec r_{1} </math> are position vector of centroids of cells cell 0 and cell 1 respectively. <br> | <math> \vec r_{0} </math> and <math> \vec r_{1} </math> are position vector of centroids of cells cell 0 and cell 1 respectively. <br> | ||

<math> {\rm{d\vec s}} = \vec r_{1} - \vec r_{0} </math> | <math> {\rm{d\vec s}} = \vec r_{1} - \vec r_{0} </math> | ||

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- | === | + | === Approach 1 === |

+ | Another approach is to use a simple expression for estimating the gradient of scalar normal to the face. <br> | ||

+ | :<math> | ||

+ | 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] | ||

+ | </math> <br> | ||

+ | where <math> \Gamma _f </math> is suitable face averages. <br> | ||

+ | |||

+ | 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. <br> | ||

+ | |||

+ | |||

+ | === Approach 2 === | ||

We define vector | We define vector | ||

<math> | <math> |

## Revision as of 01:06, 15 September 2005

## Contents |

## 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**

and are position vector of centroids of cells cell 0 and cell 1 respectively.

We wish to approaximate at the face.

### Approach 1

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

where 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

giving us the expression:

where and are suitable face averages.