4.12 Symmetry condition
The transform boundary condition was presented in Sec. 4.11 . It provides a convenient framework for implementing boundary conditions that represent a geometric constraint, including the symmetry condition.
The symmetry condition is suitable for simulations where the geometry contains a plane of symmetry and the ﬂow ﬁeld is assumed symmetric. By generating a mesh on one side of a plane of symmetry and applying the symmetry condition, the number of cells, and hence solution time, is reduced.
In the context of a wall boundary, the symmetry condition is also equivalent to slip (as opposed to the common no-slip condition).
A symmetry plane is a transform condition so when the solution variable is a scalar, it reduces to zero gradient. For a vector, e.g. , the condition is zero gradient tangential to the plane, and zero ﬁxed value normal to the plane.
When is a tensor, the boundary condition requires a more precise deﬁnition, which can also be applied to a vector . The boundary can be considered as the mean of the adjacent cell and the mirror image transformed by the reﬂective transformation tensor , i.e.
Using the notation in Sec. 4.11 , the explicit boundary value is calculated using current from Eq. (4.18 ) and the explicit gradient is calculated by Eq. (4.17 ).
Comparing Eq. (4.18) with the transform condition of Sec. 4.11 , the factor corresponds to the tensor . For a vector ﬁeld, a factor which gives good solution convergence is
For a tensor ﬁeld, good convergence is achieved with a tensor calculated as the outer product of for a vector, i.e. denoting the vector factor by , the tensor factor is .
The axes , , , introduced in Sec. 2.1 , must remain orthogonal under a transformation. It requires the transpose of the transformation tensor to equal its inverse, i.e. .
The orthogonality condition is therefore The reﬂective transformation satisﬁes the orthogonality condition since