Sec. 4.8 concluded with a table of factors for the contributions to coeﬃcients of and for the ﬁxed value and ﬁxed gradient boundary conditions.

It distinguishes between contributions for the discretisation of an advection term which requires values at faces, and the Laplacian term which requires the normal gradient.

A mixed ﬁxed value/gradient condition is deﬁned by introducing a value fraction for which

 (4.9)
Within the range the condition operates in between ﬁxed value and ﬁxed gradient. The mixed condition is simply implemented by blending the ﬁxed value and ﬁxed gradient contributions by , as shown in the table below.

This mixed condition provides the framework for a boundary condition that can switch between a ﬁxed value and ﬁxed gradient , by changing . Switching is often based on ﬂow direction, corresponding to inﬂow and to outﬂow.

Some boundary conditions can operate in the range of value fractions . The Robin condition, described next, can also be expressed as a mixed condition with varying .

### Robin condition

The Robin condition5 combines the value and normal gradient at the boundary through an expression:

 (4.10)
where is a scalar coeﬃcient with units of length and is some constant value of .

The Robin condition can be treated like the mixed condition by relating to a reference ﬁxed value and gradient at a boundary, according to .

Substituting for in Eq. (4.10) and making the subject of the equation gives:

Comparison with the value factors in the previous table shows that the Robin condition corresponds to the mixed condition with the value fraction .

In this form, and relate to the limits and , respectively. Values can be selected in these limits to represent the physics of the condition.

In many cases the reference gradient is such that in Eq. (4.10 ). For example a condition for temperature would tend to as and as .

The value fraction includes , so the condition operates “in the middle” between the ﬁxed value and gradient when is the same order of magnitude as the boundary cell height.

5after Victor Robin (1855-1897).

Notes on CFD: General Principles - 4.9 Mixed ﬁxed value/gradient