3.3 Finite 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.

PICT\relax \special {t4ht=

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

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

PICT\relax \special {t4ht=

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)eqn and face centre eqn.

Calculating mesh data

PICT\relax \special {t4ht=

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

St = (pi+1 pi) (a pi)=2 Ct = (pi+1 + pi + a)=3; \relax \special {t4ht=
where “eqn” is the cross product, Eq. (2.70 ). The sum of area vectors gives eqn and eqn is the area weighted sum of triangle centres over eqn triangles, calculated by
 N C = -1--X jS jC : f jSfj t t \relax \special {t4ht=
The cell volume eqn can be calculated from eqn and eqn by Gauss’s theorem, using eqn to describe position and noting eqn. The surface integral (eqn) becomes a sum over cell faces eqn, replacing discrete values eqn and eqn for eqn and eqn respectively as follows:
 Z 1 Z 1 Z 1X V = dV = -- r xdV = -- (dS x) -- Sf Cf: V 3 V 3 S 3 f \relax \special {t4ht=
The cell centre is calculated similarly, noting eqn, by
pict\relax \special {t4ht=
Notes on CFD: General Principles - 3.3 Finite volume mesh