254
8. Complex Geometries
In the equation for us, if it needs to be solved, one has to include the
source term (the apparent Coriolis force):
The only other difference compared to planar 2D problems is the calculation of cell face areas and volumes. The areas of cell faces 'n', 'e', 'w' and
's' are calculated as in plane geometry, see Eq. (8.13), with the inclusion of
a factor of r, (where c denotes the cell face center). The areas of the front
and back faces are calculated in the same way as the volume in plane geometry (where the third dimension is unity), see Eq. (8.39). The volume of
axi-symmetric CVs with any number of faces is:
where N, denotes the number of vertices, counted counter-clockwise, with
i = 0 corresponding to i = N,.
An important issue in axi-symmetric swirling flows is the coupling of
radial and circumferential velocity components. The equation for v, contains
vi, and the equation for vs the product of v, and vs as source terms, see above.
The combination of the sequential (decoupled) solution procedure and Picard
linearization may prove inefficient. The coupling can be improved by using
the multigrid method for the outer iterations, see Chap. 11, by a coupled
solution method, or by using more implicit linearization schemes, see Chap.
5.
If the coordinates z and r of the cylindrical coordinate system are replaced
by x and y, the analogy with the equations in Cartesian coordinates becomes
obvious. Indeed, if r is set to unity and vs and TOO are set to zero, these
equations become identical to those in Cartesian coordinates, with v, = u,
and v, = u,. Thus the same computer code can be used for both plane and
axi-symmetric 2D flows; for axi-symmetric problems, one sets r = y and
includes
and, if the swirl component is non-zero, the ve equation.
An example of application of the FV solution method described above to
axi-symmetric problems is shown in Chap. 9.
8.10 Implement ation of Boundary Conditions
The implementation of the boundary conditions on non-orthogonal grids requires special attention because the boundaries are usually not aligned with
the Cartesian velocity components. The FV method requires that the boundary fluxes either be known or expressed in terms of known quantities and
interior nodal values. Of course, the number of CVs must match the number
of unknowns.
8. Complex Geometries
In the equation for us, if it needs to be solved, one has to include the
source term (the apparent Coriolis force):
The only other difference compared to planar 2D problems is the calculation of cell face areas and volumes. The areas of cell faces 'n', 'e', 'w' and
's' are calculated as in plane geometry, see Eq. (8.13), with the inclusion of
a factor of r, (where c denotes the cell face center). The areas of the front
and back faces are calculated in the same way as the volume in plane geometry (where the third dimension is unity), see Eq. (8.39). The volume of
axi-symmetric CVs with any number of faces is:
where N, denotes the number of vertices, counted counter-clockwise, with
i = 0 corresponding to i = N,.
An important issue in axi-symmetric swirling flows is the coupling of
radial and circumferential velocity components. The equation for v, contains
vi, and the equation for vs the product of v, and vs as source terms, see above.
The combination of the sequential (decoupled) solution procedure and Picard
linearization may prove inefficient. The coupling can be improved by using
the multigrid method for the outer iterations, see Chap. 11, by a coupled
solution method, or by using more implicit linearization schemes, see Chap.
5.
If the coordinates z and r of the cylindrical coordinate system are replaced
by x and y, the analogy with the equations in Cartesian coordinates becomes
obvious. Indeed, if r is set to unity and vs and TOO are set to zero, these
equations become identical to those in Cartesian coordinates, with v, = u,
and v, = u,. Thus the same computer code can be used for both plane and
axi-symmetric 2D flows; for axi-symmetric problems, one sets r = y and
includes
and, if the swirl component is non-zero, the ve equation.
An example of application of the FV solution method described above to
axi-symmetric problems is shown in Chap. 9.
8.10 Implement ation of Boundary Conditions
The implementation of the boundary conditions on non-orthogonal grids requires special attention because the boundaries are usually not aligned with
the Cartesian velocity components. The FV method requires that the boundary fluxes either be known or expressed in terms of known quantities and
interior nodal values. Of course, the number of CVs must match the number
of unknowns.