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Diffusion term

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<br>
<br>
For a general control volume (orthogonal, non-orthogonal), the discretization of the diffusion term can be written in the following form<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>
+
 
 +
:<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  
where  
*S denotes the surface area of the control volume
*S denotes the surface area of the control volume
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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>
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>
-
<math> \overrightarrow{d_{PN}}=  \vec r_{P}  - \vec r_{N}  </math>
+
:<math> \overrightarrow{d_{PN}}=  \vec r_{P}  - \vec r_{N}  </math>
<br>
<br>
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=== Approach 2 ===
=== Approach 2 ===
We define the vector
We define the vector
-
<math>
+
:<math>
\vec \alpha {\rm{ = }}\frac{{{\rm{\vec {S_f}}}}}{{{\rm{\vec S_f}} \cdot {\overrightarrow{d_{PN}}}}}
\vec \alpha {\rm{ = }}\frac{{{\rm{\vec {S_f}}}}}{{{\rm{\vec S_f}} \cdot {\overrightarrow{d_{PN}}}}}
</math>
</math>
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=== Orthogonal correction approaches ===
=== Orthogonal correction approaches ===
In non-orthogonal grids, the gradient direction that will yield an expression involving the values at the neighboring control volumes will have to be along the line joining the centroids of the two control volumes. If this direction has a unit vector denoted by <math>\vec e</math> then, by definition <br>
In non-orthogonal grids, the gradient direction that will yield an expression involving the values at the neighboring control volumes will have to be along the line joining the centroids of the two control volumes. If this direction has a unit vector denoted by <math>\vec e</math> then, by definition <br>
-
<math>
+
:<math>
\vec e {\rm{ = }} \frac{{{\rm{\overrightarrow{d_{PN}}}}}} {\left| {\overrightarrow{d_{PN}}} \right|}
\vec e {\rm{ = }} \frac{{{\rm{\overrightarrow{d_{PN}}}}}} {\left| {\overrightarrow{d_{PN}}} \right|}
</math><br>
</math><br>
then the gradient in the direction of <math>\vec e</math> can be written as <br>
then the gradient in the direction of <math>\vec e</math> can be written as <br>
-
<math> \nabla \phi _f \cdot \vec e =  \frac {\partial \phi_f} {\partial e} = \frac { \phi_N - \phi_P} {\left| {\overrightarrow{d_{PN}}} \right|} </math> <br>
+
:<math> \nabla \phi _f \cdot \vec e =  \frac {\partial \phi_f} {\partial e} = \frac { \phi_N - \phi_P} {\left| {\overrightarrow{d_{PN}}} \right|} </math> <br>
If the surface vector <math>\vec {S_f}</math> is written as the summation of two vectors <math>\vec {E}</math> and <math>\vec {T}</math> <br>
If the surface vector <math>\vec {S_f}</math> is written as the summation of two vectors <math>\vec {E}</math> and <math>\vec {T}</math> <br>
-
<math>\vec {S_f} = \vec {E} + \vec {T}</math> <br>
+
:<math>\vec {S_f} = \vec {E} + \vec {T}</math> <br>
where <math>\vec {E}</math> is in the direction joining the centroids of the two control volumes, we will then be able to express the diffusive flux in terms of the neighboring control volumes plus an additional correction. This is done as follows <br>
where <math>\vec {E}</math> is in the direction joining the centroids of the two control volumes, we will then be able to express the diffusive flux in terms of the neighboring control volumes plus an additional correction. This is done as follows <br>
-
<math> \nabla \phi_f \cdot \vec {S_f} = \nabla \phi_f \cdot \vec {E} + \nabla \phi_f \cdot \vec {T} </math> <br>
+
:<math> \nabla \phi_f \cdot \vec {S_f} = \nabla \phi_f \cdot \vec {E} + \nabla \phi_f \cdot \vec {T} </math> <br>
-
<math> \nabla \phi_f \cdot \vec {S_f} = E \nabla \phi_f \cdot \vec {e} + \nabla \phi_f \cdot \vec {T} </math> .... (where E is the magnitude of <math> \vec E </math><br>
+
:<math> \nabla \phi_f \cdot \vec {S_f} = E \nabla \phi_f \cdot \vec {e} + \nabla \phi_f \cdot \vec {T} </math> .... (where E is the magnitude of <math> \vec E </math><br>
At the outset, one obtains <br>
At the outset, one obtains <br>
-
<math> \nabla \phi_f \cdot \vec {S_f} = E \frac { \phi_N - \phi_P} {\left| {\overrightarrow{d_{PN}}} \right|} + \nabla \phi_f \cdot \vec {T} </math> <br>
+
:<math> \nabla \phi_f \cdot \vec {S_f} = E \frac { \phi_N - \phi_P} {\left| {\overrightarrow{d_{PN}}} \right|} + \nabla \phi_f \cdot \vec {T} </math> <br>
<br>
<br>
The first term in the above equation can be thought of as the orthogonal contribution to the diffusive flux, while the second term represents the non-orthogonal effects. At this point, the vector <math>\vec {T}</math> has not been defined yet. There are three main methods to define this vector.
The first term in the above equation can be thought of as the orthogonal contribution to the diffusive flux, while the second term represents the non-orthogonal effects. At this point, the vector <math>\vec {T}</math> has not been defined yet. There are three main methods to define this vector.
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==== Minimum correction ====
==== Minimum correction ====
In the minimum correction approach, the vectors are defined as <br>
In the minimum correction approach, the vectors are defined as <br>
-
<math> \vec E = (\vec e \cdot \vec S_f)\cdot \vec e = S_f \cos\theta \vec e </math> <br>
+
:<math> \vec E = (\vec e \cdot \vec S_f)\cdot \vec e = S_f \cos\theta \vec e </math> <br>
-
<math> \vec T = \vec S - \vec E= S_f (\vec n - \cos\theta \vec e) </math> <br>
+
:<math> \vec T = \vec S - \vec E= S_f (\vec n - \cos\theta \vec e) </math> <br>
[[image:Non_orthogonal_CV_minimum_correction.jpg]] <br>
[[image:Non_orthogonal_CV_minimum_correction.jpg]] <br>
'''Minimum Correction Approach'''
'''Minimum Correction Approach'''
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==== Orthogonal correction ====
==== Orthogonal correction ====
In the orthogonal correction approach, the vectors are defined as <br>
In the orthogonal correction approach, the vectors are defined as <br>
-
<math> \vec E = S_f \vec e </math> <br>
+
:<math> \vec E = S_f \vec e </math> <br>
-
<math> \vec T = \vec S - \vec E = S_f (\vec n - \vec e) </math> <br>
+
:<math> \vec T = \vec S - \vec E = S_f (\vec n - \vec e) </math> <br>
[[image:Non_orthogonal_CV_orthogonal_correction.jpg]] <br>
[[image:Non_orthogonal_CV_orthogonal_correction.jpg]] <br>
'''Orthogonal Correction Approach'''
'''Orthogonal Correction Approach'''
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==== Over relaxed correction ====
==== Over relaxed correction ====
Finally, in the over relaxed approach, we define <br>
Finally, in the over relaxed approach, we define <br>
-
<math> \vec E = \frac {\vec S_f \cdot \vec S_f}{\vec S_f \cdot \vec e} \vec e = \frac {S_f}{\cos \theta} \vec e </math> <br>
+
:<math> \vec E = \frac {\vec S_f \cdot \vec S_f}{\vec S_f \cdot \vec e} \vec e = \frac {S_f}{\cos \theta} \vec e </math> <br>
-
<math> \vec T = \vec S - \vec E = S_f (\vec n - \frac{1}{\cos \theta}  \vec e) </math> <br>
+
:<math> \vec T = \vec S - \vec E = S_f (\vec n - \frac{1}{\cos \theta}  \vec e) </math> <br>
[[image:Non_orthogonal_CV_Over_relaxed_correction.jpg]] <br>
[[image:Non_orthogonal_CV_Over_relaxed_correction.jpg]] <br>
'''Over Relaxed Correction Approach'''
'''Over Relaxed Correction Approach'''

Revision as of 09:21, 10 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


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.

Orthogonal correction approaches

In non-orthogonal grids, the gradient direction that will yield an expression involving the values at the neighboring control volumes will have to be along the line joining the centroids of the two control volumes. If this direction has a unit vector denoted by \vec e then, by definition


\vec e {\rm{ = }} \frac{{{\rm{\overrightarrow{d_{PN}}}}}} {\left| {\overrightarrow{d_{PN}}} \right|}

then the gradient in the direction of \vec e can be written as

 \nabla \phi _f \cdot \vec e =  \frac {\partial \phi_f} {\partial e} = \frac { \phi_N - \phi_P} {\left| {\overrightarrow{d_{PN}}} \right|}

If the surface vector \vec {S_f} is written as the summation of two vectors \vec {E} and \vec {T}

\vec {S_f} = \vec {E} + \vec {T}

where \vec {E} is in the direction joining the centroids of the two control volumes, we will then be able to express the diffusive flux in terms of the neighboring control volumes plus an additional correction. This is done as follows

 \nabla \phi_f \cdot \vec {S_f} = \nabla \phi_f \cdot \vec {E} + \nabla \phi_f \cdot \vec {T}
 \nabla \phi_f \cdot \vec {S_f} = E \nabla \phi_f \cdot \vec {e} + \nabla \phi_f \cdot \vec {T} .... (where E is the magnitude of  \vec E

At the outset, one obtains

 \nabla \phi_f \cdot \vec {S_f} = E \frac { \phi_N - \phi_P} {\left| {\overrightarrow{d_{PN}}} \right|} + \nabla \phi_f \cdot \vec {T}


The first term in the above equation can be thought of as the orthogonal contribution to the diffusive flux, while the second term represents the non-orthogonal effects. At this point, the vector \vec {T} has not been defined yet. There are three main methods to define this vector.

Minimum correction

In the minimum correction approach, the vectors are defined as

 \vec E = (\vec e \cdot \vec S_f)\cdot \vec e = S_f \cos\theta \vec e
 \vec T = \vec S - \vec E= S_f (\vec n - \cos\theta \vec e)

Non orthogonal CV minimum correction.jpg
Minimum Correction Approach

Orthogonal correction

In the orthogonal correction approach, the vectors are defined as

 \vec E = S_f \vec e
 \vec T = \vec S - \vec E = S_f (\vec n - \vec e)

Non orthogonal CV orthogonal correction.jpg
Orthogonal Correction Approach

Over relaxed correction

Finally, in the over relaxed approach, we define

 \vec E = \frac {\vec S_f \cdot \vec S_f}{\vec S_f \cdot \vec e} \vec e = \frac {S_f}{\cos \theta} \vec e
 \vec T = \vec S - \vec E = S_f (\vec n - \frac{1}{\cos \theta}  \vec e)

Non orthogonal CV Over relaxed correction.jpg
Over Relaxed Correction Approach

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).
  3. Darwish, Marwan (2003), "CFD Course Notes", Notes, American University of Beirut.
  4. Saad, Tony (2005), "Implementation of a Finite Volume Unstructured CFD Solver Using Cluster Based Parallel Computing", Thesis, American University of Beirut.

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