4.9 Mixed ﬁxed value/gradient
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
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 .
The Robin condition5 combines the value and normal gradient at the boundary through an expression:
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:
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.