4.11 Transform condition

PICT\relax \special {t4ht=

Some boundary conditions represent eqn at the boundary as a transformation of the cell value eqn. They can be expressed in terms of a general transform condition

 N = X G(T ; ); f i=1 i P \relax \special {t4ht=
where eqn is the geometric transformation of variable eqn by a tensor eqn. The transformation is calculated as follows:
 8 >> for scalar ; < P G(T; P) = >> T P for vector ; : T P TT for tensor . \relax \special {t4ht=
When eqn is a scalar, the transform condition is equivalent to a zero gradient condition. Otherwise, it is implemented so that terms in eqn in Eq. (4.13 ) contribute to coefficients in eqn. These contributions are implicit which improves convergence when solving the matrix equation eqn.

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

The multiplication of each coefficient of eqn by its respective component of eqn is denoted by by eqn. In the case of vector eqn, this “component multiplication” is

P = ( x Px; y Py; z Pz): \relax \special {t4ht=
For advection discretisation, the face value is represented as
f = e |(---1---------------------{)z-----------------------P-}+ (1|------{z----)--} P; boundary, explicit internal \relax \special {t4ht=
where eqn is an explicit boundary value, calculated from the expression in Eq. (4.13 ) using the current values eqn. The other explicit term uses the current eqn with “eqn” denoting eqn for a vector eqn and eqn for a tensor.

Laplacian discretisation requires the face normal gradient eqn. Combining eqn with Eq. (4.15 ) gives

rn f = (rn |---)e-------+------C------------{z--------------------------P--} C |--{z-- } P; boundary, explicit internal \relax \special {t4ht=
where the explicit gradient is calculated by
(rn )e = C ( e P): \relax \special {t4ht=
The transform condition is summarised within the table below by the value and gradient contributions.

factor transform

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

Notes on CFD: General Principles - 4.11 Transform condition