# Diffusion term

(Difference between revisions)
 Revision as of 22:56, 14 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 01:06, 15 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 1: Line 1: ==Discretisation of Diffusive Term == ==Discretisation of Diffusive Term == - === Sub-topics === + === Description=== - ---- +
- # [[Approximation Schemes for diffusive term]] + Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes. - # [[Approximation of Diffusive Fluxes]] +
- # [[Discretisation on orthogonal grids]] + - # [[Discretisation on non-orthogonal curvilinear body-fitted grids]] + - ---- + - ===1. Description=== + 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.

[[Image:Nm_descretisation_diffusionterms_01.jpg]]
[[Image:Nm_descretisation_diffusionterms_01.jpg]]
'''Figure 1.1'''
'''Figure 1.1'''
+ :
$\vec r_{0}$ and $\vec r_{1}$ are position vector of centroids of cells cell 0 and cell 1 respectively.
$\vec r_{0}$ and $\vec r_{1}$ are position vector of centroids of cells cell 0 and cell 1 respectively.
${\rm{d\vec s}} = \vec r_{1} - \vec r_{0}$ ${\rm{d\vec s}} = \vec r_{1} - \vec r_{0}$ Line 20: Line 17: - ===2. Approach 1 === + === 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 We define vector [itex] [itex]

## 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.