## 3.3Finite volume mesh

The ﬁnite volume numerical method is closely associated with the concept of surface and volume integrals used in Chapter 2 . Below, we extract the main geometric elements from the ﬁgures used for the conservation laws.

The integrals use volumes and area vectors . The numerical method uses equivalent discrete quantities for cells and faces:

• cell volume ;
• face area vector , with area magnitude ;
• face unit normal vector .

The ﬁnite volume method relates discrete values of ﬁelds, e.g. pressure, to cells and faces within the mesh. For many calculations data must to be assigned to point locations, in particular the cell centre (more speciﬁcally centroid) and face centre .

### Calculating mesh data

To calculate , each polygonal face is decomposed into triangles using an apex point . The area vector and centre are then calculated for each triangle according to

where “” is the cross product, Eq. (2.70 ). The sum of area vectors gives and is the area weighted sum of triangle centres over triangles, calculated by The cell volume can be calculated from and by Gauss’s theorem, using to describe position and noting . The surface integral () becomes a sum over cell faces , replacing discrete values and for and respectively as follows: The cell centre is calculated similarly, noting , by
Notes on CFD: General Principles - 3.3 Finite volume mesh