8.6 Finite Volume Methods
241
approximations, which is relatively easy to do on structured grids (see Chap.
3). In FV methods, there are two approximation levels: approximation of the
integrals and approximation of the variable values at locations other than
CV center. The second order accuracy of the midpoint rule and linear interpolation is the highest accuracy achievable with single-point approximations.
Any higher order FV scheme requires interpolation of higher order at more
than one cell face location. This is manageable on structured grids, but rather
difficult on unstructured grids. For the sake of simplicity of implementation,
extension, debugging and maintenance, second order accuracy appears to be
the best compromise. Only when very high accuracy is required (discretization errors below 1%) do higher order methods become cost-effective. One
also has to bear in mind that higher order methods produce more accurate
results than a second order method only if the grid is sufficiently fine. If the
grid is not fine enough, higher order methods may produce oscillatory solutions, and the average error may be higher than for a second order scheme.
Higher order schemes also require more memory and computing time per
grid point than second order schemes. For industrial applications, for which
errors of the order of 1% are acceptable, a second order scheme coupled with
local grid refinement offers the best combination of accuracy, simplicity of
programming and code maintenance, robustness, and efficiency.
8.6.5 Block-Structured Grids
Structured grids are difficult, sometimes impossible, to construct for complex
geometries. For example, to compute the flow around a circular cylinder in a
free stream, one can easily generate a structured 0-type grid around it, but
if the cylinder is located in a narrow duct, this is no longer possible; see Fig.
2.2. In such a case, block-structured grids provide a useful compromise between the simplicity and wide variety of solvers available for structured grids
and ability to handle complex geometries that unstructured grids allow. The
idea is to use a regular data structure (lexicographic ordering) within while
constructing the blocks so as to fill the irregular domain. Many approaches
are possible. Some use overlapping blocks (e.g. Hinatsu and Ferziger, 1991;
Perng and Street, 1991; Zang and Street, 1995; Hubbard and Chen, 1994,
1995). Others rely on non-overlapping blocks (e.g. Coelho et al., 1991; Lilek
et al., 1997b). We shall describe one approach that used non-overlapping
blocks. It is also well suited for use on parallel computers (see Chap. 11);
normally, the computing for each block is assigned to a separate processor.
The solution domain is first subdivided into several sub-domains in such
a way that each sub-domain can be fitted with a structured grid with good
properties (not too non-orthogonal, individual CV aspect ratios not too
large). An example is shown in Fig. 2.2. Within each block the indices i
and j are used to identify the CVs, but we also need a block-identifier. The
data is stored in a one dimensional array. The index of the node (i, j ) in block
3 within that one-dimensional array is (see Table 3.2):
241
approximations, which is relatively easy to do on structured grids (see Chap.
3). In FV methods, there are two approximation levels: approximation of the
integrals and approximation of the variable values at locations other than
CV center. The second order accuracy of the midpoint rule and linear interpolation is the highest accuracy achievable with single-point approximations.
Any higher order FV scheme requires interpolation of higher order at more
than one cell face location. This is manageable on structured grids, but rather
difficult on unstructured grids. For the sake of simplicity of implementation,
extension, debugging and maintenance, second order accuracy appears to be
the best compromise. Only when very high accuracy is required (discretization errors below 1%) do higher order methods become cost-effective. One
also has to bear in mind that higher order methods produce more accurate
results than a second order method only if the grid is sufficiently fine. If the
grid is not fine enough, higher order methods may produce oscillatory solutions, and the average error may be higher than for a second order scheme.
Higher order schemes also require more memory and computing time per
grid point than second order schemes. For industrial applications, for which
errors of the order of 1% are acceptable, a second order scheme coupled with
local grid refinement offers the best combination of accuracy, simplicity of
programming and code maintenance, robustness, and efficiency.
8.6.5 Block-Structured Grids
Structured grids are difficult, sometimes impossible, to construct for complex
geometries. For example, to compute the flow around a circular cylinder in a
free stream, one can easily generate a structured 0-type grid around it, but
if the cylinder is located in a narrow duct, this is no longer possible; see Fig.
2.2. In such a case, block-structured grids provide a useful compromise between the simplicity and wide variety of solvers available for structured grids
and ability to handle complex geometries that unstructured grids allow. The
idea is to use a regular data structure (lexicographic ordering) within while
constructing the blocks so as to fill the irregular domain. Many approaches
are possible. Some use overlapping blocks (e.g. Hinatsu and Ferziger, 1991;
Perng and Street, 1991; Zang and Street, 1995; Hubbard and Chen, 1994,
1995). Others rely on non-overlapping blocks (e.g. Coelho et al., 1991; Lilek
et al., 1997b). We shall describe one approach that used non-overlapping
blocks. It is also well suited for use on parallel computers (see Chap. 11);
normally, the computing for each block is assigned to a separate processor.
The solution domain is first subdivided into several sub-domains in such
a way that each sub-domain can be fitted with a structured grid with good
properties (not too non-orthogonal, individual CV aspect ratios not too
large). An example is shown in Fig. 2.2. Within each block the indices i
and j are used to identify the CVs, but we also need a block-identifier. The
data is stored in a one dimensional array. The index of the node (i, j ) in block
3 within that one-dimensional array is (see Table 3.2):