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==Discretisation of Diffusive Term ==
==Discretisation of Diffusive Term ==
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=== Sub-topics ===
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=== Description===  
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----
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<br>
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# [[Approximation Schemes for diffusive term]]
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Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes.
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# [[Approximation of Diffusive Fluxes]]
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<br>
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# [[Discretisation on orthogonal grids]]
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# [[Discretisation on non-orthogonal curvilinear body-fitted grids]]
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----
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===1. Description===
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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>
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:<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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===2. Approach 1 ===
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=== Approach 1 ===
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Another approach is to use a simple expression for estimating the gradient of scalar normal to the face. <br>
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:<math>
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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]
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</math> <br>
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where <math> \Gamma _f  </math> is suitable face averages. <br>
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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>
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=== 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.
Nm descretisation diffusionterms 01.jpg
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.

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