Discretisation of the Diffusion Term
For a general control volume (orthogonal, non-orthogonal), the discretization of the diffusion term can be written in the following form
- S denotes the surface area of the control volume
- denotes the area of a face for the 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 represents the magnitude of area of the face. And n represents the normal unit vector of the face under consideration.
If and are position vector of centroids of cells P and N respectively. Then, we define
We wish to approaximate the diffusive flux at the face.
A first approach is to use a simple expression for estimating the gradient of a scalar normal to the face.
where 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.
We define the vector
giving us the expression:
where and are suitable face averages.
- Ferziger, J.H. and Peric, M. (2001), Computational Methods for Fluid Dynamics, ISBN 3540420746, 3rd Rev. Ed., Springer-Verlag, Berlin..
- 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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