378
12. Special Topics
The unsteady term has to be treated in a way that is consistent with the
space conservation law. For incompressible flows, the contribution of the grid
movement to the mass fluxes has to cancel the unsteady term, i.e. the mass
conservation equation reduces to:
The discretization method must ensure that the unsteady term and the mass
fluxes satisfy this equation if strict mass conservation is to be obtained. If the
volume change and mass fluxes are calculated as above, this conservation is
assured. Therefore, for incompressible flows, grid movement does not affect
the pressure-correction equation.
Fig. 12.3. On the calculation of a volume swept by
a cell face of a 3D CV;
shaded are surfaces common
to neighbor CVs.
In three dimensions, one has to be careful in calculating the volumes
swept by cell faces. Because the cell face edges may turn, the calculation of
the swept volume requires triangulation of the shaded surfaces in Fig. 12.3.
The volume can then be calculated using approach described in Sect. 8.6.4.
However, as the shaded surfaces are common to two CVs, one has to ensure
that they are triangulated in the same way for both CVs to assure space
conservation.
Extension to higher-order schemes is straightforward. For example, discretization of the SCL, Eq. (12.13) by the Crank-Nicolson scheme leads to
(see Eq. (12.16)):
The swept volume 60, is calculated in the same way as for the implicit Euler
scheme, but the mass flux is now calculated at the half step location:
12. Special Topics
The unsteady term has to be treated in a way that is consistent with the
space conservation law. For incompressible flows, the contribution of the grid
movement to the mass fluxes has to cancel the unsteady term, i.e. the mass
conservation equation reduces to:
The discretization method must ensure that the unsteady term and the mass
fluxes satisfy this equation if strict mass conservation is to be obtained. If the
volume change and mass fluxes are calculated as above, this conservation is
assured. Therefore, for incompressible flows, grid movement does not affect
the pressure-correction equation.
Fig. 12.3. On the calculation of a volume swept by
a cell face of a 3D CV;
shaded are surfaces common
to neighbor CVs.
In three dimensions, one has to be careful in calculating the volumes
swept by cell faces. Because the cell face edges may turn, the calculation of
the swept volume requires triangulation of the shaded surfaces in Fig. 12.3.
The volume can then be calculated using approach described in Sect. 8.6.4.
However, as the shaded surfaces are common to two CVs, one has to ensure
that they are triangulated in the same way for both CVs to assure space
conservation.
Extension to higher-order schemes is straightforward. For example, discretization of the SCL, Eq. (12.13) by the Crank-Nicolson scheme leads to
(see Eq. (12.16)):
The swept volume 60, is calculated in the same way as for the implicit Euler
scheme, but the mass flux is now calculated at the half step location: