Notes on Computational Fluid Dynamics: General Principles

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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 $\text{[math]}$ is a scalar, it reduces to zero gradient. For a vector, e.g. $\text{[math]}$ , the condition is zero gradient tangential to the plane, and zero ﬁxed value normal to the plane.

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

1 f = 2-[ P + G(I 2nfnf; P)]: \relax \special {t4ht=

(4.18)

Here, $\text{[math]}$ is the unit normal vector on the boundary face.

$PICT\relax \special {t4ht=$

Using the notation in Sec. 4.11 , the explicit boundary value $\text{[math]}$ is calculated using current $\text{[math]}$ from Eq. (4.18 ) and the explicit gradient $\text{[math]}$ is calculated by Eq. (4.17 ).

Comparing Eq. (4.18) with the transform condition of Sec. 4.11 , the factor $\text{[math]}$ corresponds to the tensor $\text{[math]}$ . For a vector ﬁeld, a factor which gives good solution convergence is

p ---- = diag( nfnf) = (jnfxj;jnfyj;jnfzj); \relax \special {t4ht=

(4.19)

where “ $\text{[math]}$ ” is the vector of the diagonal components of tensor $\text{[math]}$ .

For a tensor ﬁeld, good convergence is achieved with a tensor $\text{[math]}$ calculated as the outer product of $\text{[math]}$ for a vector, i.e. denoting the vector factor by $\text{[math]}$ , the tensor factor is $\text{[math]}$ .

Orthogonality condition

The axes $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , introduced in Sec. 2.1 , must remain orthogonal under a transformation. It requires the transpose of the transformation tensor $\text{[math]}$ to equal its inverse, i.e. $\text{[math]}$ .

The orthogonality condition is therefore $\text{[math]}$ The reﬂective transformation $\text{[math]}$ 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