8. Complex Geometries
Most flows in engineering practice involve complex geometries which are not
readily fit with Cartesian grids. Although the principles of discretization and
solution methods for algebraic systems described earlier may be used, many
modifications are required. The properties of the solution algorithm depend
on the choices of the grid and of the vector and tensor components, and
the arrangement of variables on the grid. These issues are discussed in this
chapter.
8.1 The Choice of Grid
When the geometry is regular (e.g. rectangular or circular), choosing the
grid is simple: the grid lines usually follow the coordinate directions. In complicated geometries, the choice is not at all trivial. The grid is subject to
constraints imposed by the discretization method. If the algorithm is designed for curvilinear orthogonal grids, non-orthogonal grids cannot be used;
if the CVs are required to be quadrilaterals or hexahedra, grids consisting of
triangles and tetrahedra cannot be used, etc. When the geometry is complex
and the constraints cannot be fulfilled, compromises have to be made.
8.1.1 Stepwise Approximation Using Regular Grids
The simplest approach uses orthogonal grids (Cartesian or polar-cylindrical).
In order to apply such a grid to solution domains with inclined or curved
boundaries, the boundaries have to be approximated by staircase-like steps.
This approach has been used, but it raises two kinds of problems:
0 The number of grid points (or CVs) per grid line is not constant, as it is
in a fully regular grid. This requires either indirect addressing, or special
arrays have t o be created that limit the index range on each line. The
computer code may need to be changed for each new problem.
0 The steps at the boundary introduce errors into the solution, especially
when the grid is coarse. The treatment of the boundary conditions a t stepwise walls also requires special attention.
Most flows in engineering practice involve complex geometries which are not
readily fit with Cartesian grids. Although the principles of discretization and
solution methods for algebraic systems described earlier may be used, many
modifications are required. The properties of the solution algorithm depend
on the choices of the grid and of the vector and tensor components, and
the arrangement of variables on the grid. These issues are discussed in this
chapter.
8.1 The Choice of Grid
When the geometry is regular (e.g. rectangular or circular), choosing the
grid is simple: the grid lines usually follow the coordinate directions. In complicated geometries, the choice is not at all trivial. The grid is subject to
constraints imposed by the discretization method. If the algorithm is designed for curvilinear orthogonal grids, non-orthogonal grids cannot be used;
if the CVs are required to be quadrilaterals or hexahedra, grids consisting of
triangles and tetrahedra cannot be used, etc. When the geometry is complex
and the constraints cannot be fulfilled, compromises have to be made.
8.1.1 Stepwise Approximation Using Regular Grids
The simplest approach uses orthogonal grids (Cartesian or polar-cylindrical).
In order to apply such a grid to solution domains with inclined or curved
boundaries, the boundaries have to be approximated by staircase-like steps.
This approach has been used, but it raises two kinds of problems:
0 The number of grid points (or CVs) per grid line is not constant, as it is
in a fully regular grid. This requires either indirect addressing, or special
arrays have t o be created that limit the index range on each line. The
computer code may need to be changed for each new problem.
0 The steps at the boundary introduce errors into the solution, especially
when the grid is coarse. The treatment of the boundary conditions a t stepwise walls also requires special attention.