248
8. Complex Geometries
where m is the iteration counter. It was shown in Sect. 7.5.2 that, for a 2D
uniform grid, the correction corresponds to a central difference approximation to the third derivative to the pressure multiplied by AX)^; it detects
oscillations and smoothes them out. The correction term may be small and
not fulfilling its role if Ap is too large. This can happen when unsteady
problems are solved using very small time steps, since Ap contains A R l A t ,
which is rarely the case. The correction term may be multiplied by a constant without affecting the consistency of the approximation. This approach
to pressure-velocity coupling on colocated grids was developed in early 1980s
and is usually attributed to Rhie and Chow (1983). It is widely used and is
employed in many commercial CFD codes.
Only the normal velocity component contributes to the mass flux through
a cell face. It depends on the pressure gradient in the normal direction. This
allows us to write the following expression for the normal velocity component
v, = v . n at a cell face:
Since A : is the same for all velocity components in a given CV (except near
some boundaries), one can replace A? by this quantity.
One can calculate the derivative of pressure in the direction normal to 'e'
face a t the neighboring CV centers and interpolate it to the cell face center.
Calculation of the normal derivative at the cell face directly would require a
coordinate transformation, which is the usual procedure on structured grids.
When using CVs of arbitrary shape, we would like to avoid use of coordinate transformations. Using shape functions is a possibility, but it results
in a complex pressure-correction equation. The deferred-correction approach
could be used to reduce the complexity.
Another approach can be constructed. From Fig. 8.9, we see that the pressure derivative with respect to n can be approximated as a central difference:
The locations of the auxiliary nodes P' and E' are easily found:
r p , = re - [(re - r p ) . n] n ; r ~ ,
= re - [(re - TE) . n] n .
(8.53)
These expressions are valid for a CV of any shape. The values of pressure
at the two auxiliary nodes can be calculated using cell-center values and
gradients:
With these expressions, Eq. (8.52) becomes:
8. Complex Geometries
where m is the iteration counter. It was shown in Sect. 7.5.2 that, for a 2D
uniform grid, the correction corresponds to a central difference approximation to the third derivative to the pressure multiplied by AX)^; it detects
oscillations and smoothes them out. The correction term may be small and
not fulfilling its role if Ap is too large. This can happen when unsteady
problems are solved using very small time steps, since Ap contains A R l A t ,
which is rarely the case. The correction term may be multiplied by a constant without affecting the consistency of the approximation. This approach
to pressure-velocity coupling on colocated grids was developed in early 1980s
and is usually attributed to Rhie and Chow (1983). It is widely used and is
employed in many commercial CFD codes.
Only the normal velocity component contributes to the mass flux through
a cell face. It depends on the pressure gradient in the normal direction. This
allows us to write the following expression for the normal velocity component
v, = v . n at a cell face:
Since A : is the same for all velocity components in a given CV (except near
some boundaries), one can replace A? by this quantity.
One can calculate the derivative of pressure in the direction normal to 'e'
face a t the neighboring CV centers and interpolate it to the cell face center.
Calculation of the normal derivative at the cell face directly would require a
coordinate transformation, which is the usual procedure on structured grids.
When using CVs of arbitrary shape, we would like to avoid use of coordinate transformations. Using shape functions is a possibility, but it results
in a complex pressure-correction equation. The deferred-correction approach
could be used to reduce the complexity.
Another approach can be constructed. From Fig. 8.9, we see that the pressure derivative with respect to n can be approximated as a central difference:
The locations of the auxiliary nodes P' and E' are easily found:
r p , = re - [(re - r p ) . n] n ; r ~ ,
= re - [(re - TE) . n] n .
(8.53)
These expressions are valid for a CV of any shape. The values of pressure
at the two auxiliary nodes can be calculated using cell-center values and
gradients:
With these expressions, Eq. (8.52) becomes: