4.13 Axisymmetric (wedge) condition

PICT\relax \special {t4ht=

There are some fluid flow problems for which the geometry is axisymmetric. Assuming the flow solution is axisymmetric, i.e. fields do not change in the circumferential direction, the computational mesh can be formed of a wedge-shaped slice of the flow geometry.

This type of mesh for axisymmetric solution contains one cell across the circumferential direction, which reduces the number of cells to two dimensions in the axial and radial directions.

This approach to axisymmetric solution introduces a geometric error due the faces normal to the radial direction being flat. This error reduces with decreasing wedge angle; in practice, the error can be considered negligible for an angle of 1eqn.

The wedge boundary condition is applied to the two sloping side patches. It transforms cell values eqn to the patch faces using a rotational transformation tensor eqn by

f = G(Rf; P): \relax \special {t4ht=
eqn defines a rotation between the unit vector eqn in the circumferential direction at the cell centre and the unit face normal vector eqn by eqn

PICT\relax \special {t4ht=

The wedge condition uses the general transform framework of Sec. 4.11 , with the explicit value eqn calculated using current eqn from Eq. (4.20). The explicit gradient eqn is the boundary gradient eqn calculated from eqn in an imaginary neighbour cell by

 C--- rn b = 2 [G(RN; P) P]; \relax \special {t4ht=
where the rotation between cell centres eqn.

The factor eqn is chosen to minimise the gradient boundary coefficients (see Sec. 4.11 ). The vector factor is eqn where “eqn” is defined in Sec. 4.12 .

For a tensor eqn, the factor is eqn, where eqn are the coefficients for a vector, as described for the symmetry condition in Sec. 4.12 .

Rotation tensor

The rotation tensor eqn between two unit vectors eqn and eqn can be calculated using the Euler-Rodrigues rotation formula,6

R = cI+ (1 c)kk + n2n1 n1n2; \relax \special {t4ht=
where eqn and eqn.
6Benjamin Olinde Rodrigues, De l’attraction des sphéroïdes, 1815.

Notes on CFD: General Principles - 4.13 Axisymmetric (wedge) condition