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.

PICT\relax \special {t4ht=

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 eqn is a scalar, it reduces to zero gradient. For a vector, e.g. eqn, the condition is zero gradient tangential to the plane, and zero fixed value normal to the plane.

When eqn is a tensor, the boundary condition requires a more precise definition, which can also be applied to a vector eqn. The boundary eqn can be considered as the mean of the adjacent cell eqn and the mirror image eqn transformed by the reflective transformation tensor eqn, i.e.

 1 f = 2-[ P + G(I 2nfnf; P)]: \relax \special {t4ht=
Here, eqn is the unit normal vector on the boundary face.

PICT\relax \special {t4ht=

Using the notation in Sec. 4.11 , the explicit boundary value eqn is calculated using current eqn from Eq. (4.18 ) and the explicit gradient eqn is calculated by Eq. (4.17 ).

Comparing Eq. (4.18) with the transform condition of Sec. 4.11 , the factor eqn corresponds to the tensor eqn. For a vector field, a factor which gives good solution convergence is

 p ---- = diag( nfnf) = (jnfxj;jnfyj;jnfzj); \relax \special {t4ht=
where “eqn” is the vector of the diagonal components of tensor eqn.

For a tensor field, good convergence is achieved with a tensor eqn calculated as the outer product of eqn for a vector, i.e. denoting the vector factor by eqn, the tensor factor is eqn.

Orthogonality condition

The axes eqn, eqn, eqn, introduced in Sec. 2.1 , must remain orthogonal under a transformation. It requires the transpose of the transformation tensor eqn to equal its inverse, i.e. eqn.

The orthogonality condition is therefore eqn The reflective transformation eqn satisfies the orthogonality condition since

T TT = (I 2nfnf) (I 2nfnf) = I 4n n + 4n n n n = I: f f f f f f \relax \special {t4ht=
Notes on CFD: General Principles - 4.12 Symmetry condition