## 4.11Transform condition Some boundary conditions represent at the boundary as a transformation of the cell value . They can be expressed in terms of a general transform condition (4.13)
where is the geometric transformation of variable by a tensor . The transformation is calculated as follows: (4.14)
When is a scalar, the transform condition is equivalent to a zero gradient condition. Otherwise, it is implemented so that terms in in Eq. (4.13 ) contribute to coeﬃcients in . These contributions are implicit which improves convergence when solving the matrix equation .

A factor is introduced to specify the contribution to internal coeﬃcients. It represents a single internal coeﬃcient for each component of so has the same rank as , i.e. it is a vector when is a vector, and a tensor when is a tensor.

The multiplication of each coeﬃcient of by its respective component of is denoted by by . In the case of vector , this “component multiplication” is For advection discretisation, the face value is represented as (4.15)
where is an explicit boundary value, calculated from the expression in Eq. (4.13 ) using the current values . The other explicit term uses the current with “ ” denoting for a vector and for a tensor.

Laplacian discretisation requires the face normal gradient . Combining with Eq. (4.15 ) gives (4.16)
where the explicit gradient is calculated by (4.17)
The transform condition is summarised within the table below by the value and gradient contributions.

 factor transform value internal value boundary gradient internal gradient boundary Notes on CFD: General Principles - 4.11 Transform condition 