220
8. Complex Geometries
either side of it that matters. Thus, a 2D grid made of equilateral triangles
is equivalent to an orthogonal grid, since lines connecting cell centers are
orthogonal to cell faces. This will be discussed further in Sect. 8.6.2.
Cell topology is also important. If the midpoint rule integral approximation, linear interpolation, and central differences are used to discretize the
equations, then the accuracy will be higher if the CVs are quadrilaterals in
2D and hexahedra in 3D, than if we use triangles and tetrahedra, respectively. The reason is that parts of the errors made at opposite cell faces when
discretizing diffusion terms cancel partially (if cell faces are parallel and of
equal area, they cancel completely) on quadrilateral and hexahedral CVs.
To obtain the same accuracy on triangles and tetrahedra, more sophisticated
interpolation and difference approximations must be used. Especially near
solid boundaries it is desirable t o have quadrilaterals or hexahedra, since all
quantities vary substantially there and accuracy is especially important in
this region.
Accuracy is also improved if one set of grid lines closely follows the streamlines of the flow, especially for the convective terms. This cannot be achieved
if triangles and/or tetrahedra are used, but is possible with quadrilaterals
and hexahedra.
Non-uniform grids are the rule rather than exception when complex geometries are treated. The ratio of the sizes of adjacent cells should be kept
under control, as accuracy is adversely affected if it is large. Especially when
block-structured grids are used, one should take care that the cells are of
nearly equal size near block interfaces; a factor of two variation should be the
maximum. An experienced user may know where strong variation of velocity,
pressure, temperature, etc. can be expected; the grid should be fine in these
regions since the errors are most likely to be large there. However, even an
experienced user will encounter occasional surprises and more sophisticated
methods are useful in any event. Errors are convected and diffused across the
domain, as discussed in Sect. 3.9, making it essential to achieve as uniform
a distribution of truncation error as possible. It is possible, however, to start
with a coarse grid and later refine it locally according to an estimate of the
discretization error; methods for doing this are called solution adaptive grid
methods and will be described in Chap. 11.
Finally, there is the issue of grid generation. When the geometry is complex, this task usually consumes the largest amount of user time by far; it is
not unusual for a designer to spend several weeks generating a single grid.
Since the accuracy of the solution depends as much (if not more) on the grid
quality as on the approximations used for discretization of the equations, grid
optimization is a worthwhile investment of time.
Many commercial codes for grid generation exist. Automation of the grid
generation process, aimed at reducing the user time and speeding up the process is the major goal in this area. Overlapping grids are easier to generate,
but there are geometries in which application of this approach is difficult
8. Complex Geometries
either side of it that matters. Thus, a 2D grid made of equilateral triangles
is equivalent to an orthogonal grid, since lines connecting cell centers are
orthogonal to cell faces. This will be discussed further in Sect. 8.6.2.
Cell topology is also important. If the midpoint rule integral approximation, linear interpolation, and central differences are used to discretize the
equations, then the accuracy will be higher if the CVs are quadrilaterals in
2D and hexahedra in 3D, than if we use triangles and tetrahedra, respectively. The reason is that parts of the errors made at opposite cell faces when
discretizing diffusion terms cancel partially (if cell faces are parallel and of
equal area, they cancel completely) on quadrilateral and hexahedral CVs.
To obtain the same accuracy on triangles and tetrahedra, more sophisticated
interpolation and difference approximations must be used. Especially near
solid boundaries it is desirable t o have quadrilaterals or hexahedra, since all
quantities vary substantially there and accuracy is especially important in
this region.
Accuracy is also improved if one set of grid lines closely follows the streamlines of the flow, especially for the convective terms. This cannot be achieved
if triangles and/or tetrahedra are used, but is possible with quadrilaterals
and hexahedra.
Non-uniform grids are the rule rather than exception when complex geometries are treated. The ratio of the sizes of adjacent cells should be kept
under control, as accuracy is adversely affected if it is large. Especially when
block-structured grids are used, one should take care that the cells are of
nearly equal size near block interfaces; a factor of two variation should be the
maximum. An experienced user may know where strong variation of velocity,
pressure, temperature, etc. can be expected; the grid should be fine in these
regions since the errors are most likely to be large there. However, even an
experienced user will encounter occasional surprises and more sophisticated
methods are useful in any event. Errors are convected and diffused across the
domain, as discussed in Sect. 3.9, making it essential to achieve as uniform
a distribution of truncation error as possible. It is possible, however, to start
with a coarse grid and later refine it locally according to an estimate of the
discretization error; methods for doing this are called solution adaptive grid
methods and will be described in Chap. 11.
Finally, there is the issue of grid generation. When the geometry is complex, this task usually consumes the largest amount of user time by far; it is
not unusual for a designer to spend several weeks generating a single grid.
Since the accuracy of the solution depends as much (if not more) on the grid
quality as on the approximations used for discretization of the equations, grid
optimization is a worthwhile investment of time.
Many commercial codes for grid generation exist. Automation of the grid
generation process, aimed at reducing the user time and speeding up the process is the major goal in this area. Overlapping grids are easier to generate,
but there are geometries in which application of this approach is difficult