4.9 Mixed fixed value/gradient

Sec. 4.8 concluded with a table of factors for the contributions to coefficients of eqn and eqn 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 eqn for which

 (0 represents fixed gradient, v = 1 represents fixed value. \relax \special {t4ht=
Within the range eqn 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 eqn, as shown in the table below.

factor mixed

value internal eqn
value boundary eqn
gradient internal eqn
gradient boundary eqn

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

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

Robin condition

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

+ arn = 0; \relax \special {t4ht=
where eqn is a scalar coefficient with units of length and eqn is some constant value of eqn.

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

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

= ---1---- b +----a---rn b + -aC----- P 1+ aC 1 + aC 1+ aC \relax \special {t4ht=
Comparison with the value factors in the previous table shows that the Robin condition corresponds to the mixed condition with the value fraction eqn .

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

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

The value fraction eqn includes eqn, so the condition operates “in the middle” between the fixed value and gradient when eqn 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