Environmental Engineering Reference
In-Depth Information
cells than can be accommodated on the computer. An
appropriate spatial scale must be selected. Furthermore,
a mesh should be refined in certain key areas such as
boundary layers where velocity gradients are highest. Ini-
tially, CFD software required the use of structured grids
where the grids were laid out in a regular repeating pat-
tern, which was often called a
block
. In 2D, the block
consisted of quadrilaterals, in 3D it was hexahedra. The
skilled user could create grids around complex geome-
tries using stretched and twisted combinations of these
basic blocks (Figure 6.1a). While the grids generated in
this way were generally of high quality and controllable,
they were time consuming to create. To overcome this
problem, unstructured grids were introduced in which
an arbitrary combination of cells fill the domain. There
is no discernable pattern to these meshes, which are con-
structed from triangles and quadrilaterals in 2D and from
tetrahedra in 3D (Figure 6.1b). The current versions of
commercial CFD software are based around unstructured
solvers which do not differentiate between structured or
unstructured grids.
The error between the solution of the partial differential
equations and their discrete representation is related
to the cell size. A method where the error reduces in
proportion to the cell size is called first order and one
where the error reduces in proportion to the square of
the cell size is called second order. With any solution
the sensitivity to the reduction in grid cell size should
be examined. It is not always possible to reduce the cell
size to a level where the solution becomes independent
of grid size, but the sensitivity should still be ascertained
in order to give confidence in the solution. Further ways
of using successively refined grids to estimate errors
are given by Lane
et al
. (2005). A further complication
when working in the natural environment is that refining
the grid gives a more refined sampling of the boundary
representation (such as the floodplain topography), which
changes the shape of the domain and thereby means that
a different problem from that on the unrefined grid is
being solved.
6.2.4 Discretizationandsolutionmethods
As described above, there are several methodologies for
converting the continuous equations to a set of alge-
braic equations for the discrete values. Even within
each methodology there are different ways of making
the approximations. These different approximations for
producing discrete values are known as discretization
techniques. Within the finite volume framework, the
equations are integrated over a cell and the problem is
reduced to finding a face value from values in surround-
ing volumes. A popular method to do so is the upwind
method (Versteeg andMalalasekera, 2007), whichuses the
intuitive idea that the face values are more closely related
to upstream rather than downstream values and therefore
approximates them by the value in the cell immediately
upstream for the face value. This scheme is the default
in some packages, but should be replaced by higher
order schemes whenever possible. All these higher order
schemes aim to reduce the error in the approximation
and hence reduce any numerically generated diffusion.
Figure 6.1
(a) Structured and (b) unstructured grid.
(a)
(b)
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