3.8 Surface normal gradient

The surface normal gradient eqn is a part of the Laplacian discretisation Eq. (3.2 ), illustrated in the figure below.

PICT\relax \special {t4ht=

The discretisation of eqn is built upon a finite difference of cell values on each side of the face according to

rn f = C ( N P); \relax \special {t4ht=
where eqn. When this orthogonal scheme is applied to Eq. (3.2 ) to discretise a Laplacian, it forms coefficients eqn of a matrix equation eqn since it references cell values of the field eqn. For cell eqn, the coefficient for each neighbour cell (eqn) is eqn and the diagonal coefficient is the negative of the sum of neighbour coefficients: eqn.

Discretisation of eqn by Eq. (3.5) is most accurate when the face is orthogonal to eqn, i.e. the angle eqn between eqn and eqn is zero. However, if the face is non-orthogonal, the error associated with Eq. (3.5 ) increases with eqn.

Non-orthogonal correction

A more accurate discretisation of eqn at a non-orthogonal face is formed of the vector sum of the orthogonal scheme eqn and an explicit correction eqn. The latter is calculated from the full gradient eqn in adjacent cells (described in Sec. 3.15 ), interpolated to the face eqn.

PICT\relax \special {t4ht=

The correction eqn is explicit, i.e. calculated using known values of eqn, so may need updating within an iterative sequence to maintain accuracy, as discussed in Sec. 5.20 . To ensure that the iterative sequence converges, the implicit contribution is elevated by replacing eqn in the orthogonal scheme with

 corr ---1--- -----1------ C = n d = j dj cos no: \relax \special {t4ht=
The corrected eqn scheme combines the implicit and explicit parts by
r = C corr( )+ (n C corr d) (r ) : n f |--------------------{Nz--------------P------} |-----------------------------{z-----------------------------}f orthogonal, implicit correction, explicit \relax \special {t4ht=
The corrected scheme is generally stable for eqn. For eqn, stability can be maintained at the expense of accuracy by limiting the magnitude of the correction eqn below some fraction of the magnitude of the orthogonal eqn part.
Notes on CFD: General Principles - 3.8 Surface normal gradient