3.19 Calculated derivatives

The introduction to matrix construction, Sec. 3.5 , described implicit and explicit discretisation of terms in an equation. It concluded that the principal derivatives of eqn that can be treated implicitly — forming matrix coefficients in eqn — are a time derivative, advection and Laplacian.

Terms with other derivatives must be calculated from respective fields, e.g. eqn from current values of eqn. In Sec. 3.15 , we have described discretisation of a gradient, which is always explicit. This section gathers together the other derivatives found in equations for fluid dynamics and associated models.

General divergence term

A general divergence term is any term that can be represented by eqn. It excludes the Laplacian term which includes a gradient eqn, and advection which includes eqn.

PICT\relax \special {t4ht=

The discretisation of a general divergence term is an explicit calculation using current values of eqn. It is based on a surface integral using the divergence definition in Sec. 2.23 as shown below:

 Z -1-- 1-X r = liVm!0 V S(dS ) ! V Sf f: f \relax \special {t4ht=
The face value eqn is generally interpolated from cell values using the linear scheme. Terms discretised using this scheme include eqn in Eq. (2.45 ), a divergence of stress eqn, etc.

Curl of a vector

The curl derivative eqn is calculated from the gradient eqn and applying the Hodge dual operator given by Eq. (2.40 ) using the following relation:

r u = 2 (skw ru) : \relax \special {t4ht=
In other words, eqn is discretised according to a scheme from Sec. 3.15 , from which eqn is calculated by Eq. (3.31 ).

Mag-square grad-grad

A derivative which appears in some model equations is eqn, described as “mag-square grad-grad”. This derivative returns a scalar since the mag-square, e.g. eqn, represents the inner product of eqn with itself, as shown in Eq. (2.7 ).

The mag-square calculation always uses the appropriate inner product to reduce the result to a scalar. For a tensor eqn, it is the double inner product, i.e. eqn.

The grad-grad operator eqn yields a third-rank tensor in the case that eqn is a vector field. To avoid storing third-rank tensors, the mag-square grad-grad operator is evaluated by summing the result from the operator on each component eqn of eqn by

N X 1 jrr ij2 ; i=0 \relax \special {t4ht=
where eqn is the number of components in eqn.
Notes on CFD: General Principles - 3.19 Calculated derivatives