258
8. Complex Geometries
implicitly and add the difference between the implicit approximation and the
force calculated using one of the above mentioned approaches to the right
hand side of the equation. Here ( is the local coordinate along the grid line
connecting cell face center and node P. The coefficient Ap is then the same
for all velocity components, and the explicit terms partially cancel out, as
discussed in Sect. 7.1. The rate of convergence is almost unaffected.
8.10.4 Symmetry Planes
In many flows there are one or more symmetry planes. When the flow is
steady, there is a solution which is symmetric with respect to this plane (in
many cases, e.g. diffusers or channels with sudden expansions, there exist
also asymmetric steady solutions). The symmetric solution can be obtained
by solving the problem in part of the solution domain only using symmetry
conditions.
At a symmetry plane the convective fluxes of all quantities are zero. Also,
the normal gradients of the velocity components parallel to symmetry plane
and of all scalar quantities are zero there. The normal velocity component is
zero, but its normal gradient is not; thus, the normal stress rnn is non-zero.
The surface integral of rnn results in a force:
When the symmetry boundary does not coincide with a Cartesian coordinate plane, the diffusive fluxes of all three Cartesian velocity components
will be non-zero. These fluxes can be calculated by obtaining first the resultant normal force from (8.81) and an approximation of the normal derivative
as described in the preceding section, and splitting this force into Cartesian
components. Alternatively, one can extrapolate the velocity gradients from
interior to the boundary and use an expression similar to (8.79), e.g. for the
u, component at the face 's' (see Fig. 8.15):
As in the case of wall boundaries, one can split the diffusive fluxes a t a
symmetry boundary into an implicit part, involving velocity component a t
the CV center (which contributes to the coefficient Ap) or use the deferredcorrection approach to keep Ap same for all velocity components.
8.10.5 Specified Pressure
In incompressible flows one usually specifies the mass flow rate a t inlet and
uses extrapolation at the outlet. However, there are situations in which the
8. Complex Geometries
implicitly and add the difference between the implicit approximation and the
force calculated using one of the above mentioned approaches to the right
hand side of the equation. Here ( is the local coordinate along the grid line
connecting cell face center and node P. The coefficient Ap is then the same
for all velocity components, and the explicit terms partially cancel out, as
discussed in Sect. 7.1. The rate of convergence is almost unaffected.
8.10.4 Symmetry Planes
In many flows there are one or more symmetry planes. When the flow is
steady, there is a solution which is symmetric with respect to this plane (in
many cases, e.g. diffusers or channels with sudden expansions, there exist
also asymmetric steady solutions). The symmetric solution can be obtained
by solving the problem in part of the solution domain only using symmetry
conditions.
At a symmetry plane the convective fluxes of all quantities are zero. Also,
the normal gradients of the velocity components parallel to symmetry plane
and of all scalar quantities are zero there. The normal velocity component is
zero, but its normal gradient is not; thus, the normal stress rnn is non-zero.
The surface integral of rnn results in a force:
When the symmetry boundary does not coincide with a Cartesian coordinate plane, the diffusive fluxes of all three Cartesian velocity components
will be non-zero. These fluxes can be calculated by obtaining first the resultant normal force from (8.81) and an approximation of the normal derivative
as described in the preceding section, and splitting this force into Cartesian
components. Alternatively, one can extrapolate the velocity gradients from
interior to the boundary and use an expression similar to (8.79), e.g. for the
u, component at the face 's' (see Fig. 8.15):
As in the case of wall boundaries, one can split the diffusive fluxes a t a
symmetry boundary into an implicit part, involving velocity component a t
the CV center (which contributes to the coefficient Ap) or use the deferredcorrection approach to keep Ap same for all velocity components.
8.10.5 Specified Pressure
In incompressible flows one usually specifies the mass flow rate a t inlet and
uses extrapolation at the outlet. However, there are situations in which the