3.6 Overview of discretisation

The choice of numerical method determines how coefficients for eqn and eqn are calculated and consequently the characteristics of the resulting matrix equation eqn.

The finite volume method described here is firmly rooted in the underlying concept of control volumes, described in Sec. 3.1 . It uses integrals over a surface surrounding a volume applied to irregular polyhedral meshes, described in Sec. 3.2 .

The discretisation is described in terms of differential operators, e.g. eqn and eqn, applied to a general field eqn.4

PICT\relax \special {t4ht=

The main concept is that faces within a mesh form closed surfaces surrounding finite volumes, e.g. a single cell. Any surface integral that represents a derivative, e.g. eqn, is approximated by a summation over the faces ‘eqn’ that form the surface, i.e.

Z X S(dS ) ! Sf f: f \relax \special {t4ht=
The flux associated with eqn, i.e. eqn, must then be calculated. The value eqn is required at each face, which must be calculated by some method of interpolation of values of eqn from cells neighbouring the respective face.

Intensive and extensive properties

In this chapter, derivatives and their discretisation are described at a point, e.g. eqn, e.g. Eq. (3.8 ):

 X r (u ) ! 1- f f: V f \relax \special {t4ht=
Calculating a derivative with this expression using the known field eqn simply produces another field (with values defined at cell centres, with units of eqn/time).

The resulting field is eqn, meaning it is independent of the size of the system/geometry. Like other intensive fields, e.g. eqn and eqn themselves, it can be used in further calculations, e.g. within another derivative or added/subtracted from other fields.

Extensive properties are dependent on the size of the system. For example, the volumetric flux eqn described in Sec. 3.9 is dependent on the face areas eqn. Numerical operations involving extensive properties, e.g. addition, subtraction, or mapping to another location, generally produce meaningless data.

While calculations of derivatives yield intensive fields, matrix equations are constructed in extensive form, with coefficients and source vector scaled by cell volume eqn. In other words, in the discretisation example above, the multiplication by eqn would be omitted.

There is no eqn multiplier in the discretisation of terms which do not involve a surface integral, e.g. time derivative Eq. (3.21 ) and terms in Sec. 3.20 . For those terms, the calculation of matrix coefficients and sources includes a multiplication by eqn.

4consistent with the idea of “Field Operation and Manipulation”, which forms the acronym “FOAM” (later, OpenFOAM) used for the CFD software created by author Henry Weller.

Notes on CFD: General Principles - 3.6 Overview of discretisation