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Diffusion term

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Discretisation of Diffusive Term


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.


  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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