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Diffusion term

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=== Description===  
=== Description===  
<br>
<br>
 +
For a general control volume (orthogonal, non-orthogonal), the discretization of the diffusion term can be written in the following form<br>
 +
<math> \int_{S}\Gamma\nabla\phi\cdot{\rm{d\vec S}}  = \sum_{faces}\Gamma _f \nabla \phi _f  \cdot{\rm{\vec S_f}} </math> <br>
 +
where
 +
*S denotes the surface area of the control volume
 +
*<math>S_f</math> 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<br>
 +
[[Image:non_orthogonal_CV_terminology.jpg]] <br>
 +
'''A general non-orthogonal control volume''' <br>
 +
Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes.
Note: The approaches those are discussed here are applicable to non-orthoganal meshes as well as orthogonal meshes.
<br>
<br>
-
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. Where '''<math>S_f</math>''' represents the magnitude of area of the face. And '''n''' represents the normal unit vector of the face under consideration.  
-
<br>
+
 
-
[[Image:Nm_descretisation_diffusionterms_01.jpg]] <br>
+
If <math> \vec r_{P} </math> and <math> \vec r_{N} </math> are position vector of centroids of cells P and N respectively. Then, we define <br>
-
'''Figure 1.1''' <br>
+
<math> \overrightarrow{d_{PN}}=  \vec r_{P}  - \vec r_{N}  </math>
-
:<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>
+
<br>
<br>
-
We wish to approaximate <math> D_f  = \Gamma _f \nabla \phi _f  \bullet {\rm{\vec A}} </math> at the face.
+
We wish to approaximate the diffusive flux <math> D_f  = \Gamma _f \nabla \phi _f  \cdot{\rm{\vec S_f}} </math> at the face.
=== Approach 1 ===
=== Approach 1 ===
-
Another approach is to use a simple expression for estimating the gradient of scalar normal to the face. <br>
+
A first approach is to use a simple expression for estimating the gradient of a scalar normal to the face. <br>
:<math>
:<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]
+
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]
</math> <br>
</math> <br>
-
where <math> \Gamma _f  </math> is suitable face averages. <br>
+
where <math> \Gamma _f  </math> is a suitable face average. <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>
+
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. <br>
=== Approach 2 ===
=== Approach 2 ===
-
We define vector
+
We define the vector
<math>
<math>
-
\vec \alpha {\rm{ = }}\frac{{{\rm{\vec A}}}}{{{\rm{\vec A}} \bullet {\rm{d\vec s}}}}
+
\vec \alpha {\rm{ = }}\frac{{{\rm{\vec {S_f}}}}}{{{\rm{\vec S_f}} \cdot {\overrightarrow{d_{PN}}}}}
</math>
</math>
giving us the expression: <br>
giving us the expression: <br>
:<math>
:<math>
-
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]
+
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]
</math> <br>
</math> <br>
where <math> \bar \nabla \phi _f  </math> and <math> \Gamma _f  </math> are suitable face averages. <br>
where <math> \bar \nabla \phi _f  </math> and <math> \Gamma _f  </math> are suitable face averages. <br>

Revision as of 04:36, 5 December 2005

Contents

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
Non orthogonal CV terminology.jpg
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).


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