8.4 The Choice of Variable Arrangement
225
components are used but there are more terms to be approximated. The conservation equations in terms of Cartesian components were given in Chap.
1.
If the FD method is used, one has only to employ the appropriate forms
of the divergence and gradient operators for non-orthogonal coordinates (or
to transform all derivatives with respect to Cartesian coordinates to the nonorthogonal coordinates). This leads to an increased number of terms, but
the conservation properties of the equations remain the same as in Cartesian
coordinates, as will be shown below.
In FV methods there is no need for coordinate transformations. When
the gradient normal to the CV surface is approximated, one can use a local
coordinate transformation, as will be shown below.
8.4 The Choice of Variable Arrangement
In Chap. 7 we mentioned that apart from colocated variable arrangement,
various staggered arrangements are possible. While there were no obvious
advantages for one or the other for Cartesian grids, the situation changes
substantially when non-orthogonal grids are used.
8.4.1 Staggered Arrangements
The staggered arrangement, presented in Chap. 7 for Cartesian grids, is applicable to non-orthogonal grids only if the grid-oriented velocity components
are employed. In Fig. 8.5 portions of such a grid are shown, in which the
grid lines change direction by 90". In one case, the contravariant, and in the
other case, the Cartesian, velocity components are shown at the staggered
locations. Recall that the staggered arrangement was introduced in order to
achieve strong coupling between the velocities and the pressure gradient. The
goal was to have the velocity component normal t o cell face lie between the
pressure nodes on either side of that face, see Fig. 7.4. For contravariant or covariant grid-oriented components, this goal is also achieved on non-orthogonal
grids, see Fig. 8.5 (a). For Cartesian components, when the grid lines change
direction by 90" a situation like the one shown in Fig. 8.5 (b) arises: the
velocity component stored a t the cell face makes no contribution to the mass
flux through that face, as it is parallel to the face. In order to calculate mass
fluxes through such CV faces, one has to use interpolated velocities from
surrounding cell faces. This makes the derivation of the pressure-correction
equation difficult, and does not ensure the proper coupling of velocities and
pressure - oscillations in either may result.
Since, in engineering flows, grid lines often change direction by 180" or
more, especially if unstructured grids are used, the staggered arrangement
is difficult to use. Some of these problems can be overcome if all Cartesian
225
components are used but there are more terms to be approximated. The conservation equations in terms of Cartesian components were given in Chap.
1.
If the FD method is used, one has only to employ the appropriate forms
of the divergence and gradient operators for non-orthogonal coordinates (or
to transform all derivatives with respect to Cartesian coordinates to the nonorthogonal coordinates). This leads to an increased number of terms, but
the conservation properties of the equations remain the same as in Cartesian
coordinates, as will be shown below.
In FV methods there is no need for coordinate transformations. When
the gradient normal to the CV surface is approximated, one can use a local
coordinate transformation, as will be shown below.
8.4 The Choice of Variable Arrangement
In Chap. 7 we mentioned that apart from colocated variable arrangement,
various staggered arrangements are possible. While there were no obvious
advantages for one or the other for Cartesian grids, the situation changes
substantially when non-orthogonal grids are used.
8.4.1 Staggered Arrangements
The staggered arrangement, presented in Chap. 7 for Cartesian grids, is applicable to non-orthogonal grids only if the grid-oriented velocity components
are employed. In Fig. 8.5 portions of such a grid are shown, in which the
grid lines change direction by 90". In one case, the contravariant, and in the
other case, the Cartesian, velocity components are shown at the staggered
locations. Recall that the staggered arrangement was introduced in order to
achieve strong coupling between the velocities and the pressure gradient. The
goal was to have the velocity component normal t o cell face lie between the
pressure nodes on either side of that face, see Fig. 7.4. For contravariant or covariant grid-oriented components, this goal is also achieved on non-orthogonal
grids, see Fig. 8.5 (a). For Cartesian components, when the grid lines change
direction by 90" a situation like the one shown in Fig. 8.5 (b) arises: the
velocity component stored a t the cell face makes no contribution to the mass
flux through that face, as it is parallel to the face. In order to calculate mass
fluxes through such CV faces, one has to use interpolated velocities from
surrounding cell faces. This makes the derivation of the pressure-correction
equation difficult, and does not ensure the proper coupling of velocities and
pressure - oscillations in either may result.
Since, in engineering flows, grid lines often change direction by 180" or
more, especially if unstructured grids are used, the staggered arrangement
is difficult to use. Some of these problems can be overcome if all Cartesian