Page 171 - Computational Fluid Dynamics for Engineers

158 5. Numerical Methods for Model Hyperbolic Equations

(c) i (d)

Fig. 5.2. Two-dimensional finite-volume mesh systems, (a) Cell centered structured finite-
volume mesh; (b) cell vertex structured finite-volume mesh; (c) cell centered unstructured
finite-volume mesh; (d) cell vertex unstructured finite-volume mesh.

kinds of meshes corresponding to structured and unstructured meshes, shown
in Fig. 5.2, can be used for this purpose. The structured meshes refer to the
type of meshes (Fig. 5.2a,b) where all mesh points lie on the intersection of
two (or three) families of lines. The unstructured meshes are those formed by
combinations of triangular (Fig. 5.2c,d) and quadrilateral cells (or tetrahedra
and pyramids in three dimensions) and the mesh points cannot be identified with
coordinate lines. Therefore, they cannot be represented by a set of integers such
as i, j but must be numbered individually in a certain order. For this reason, the
use of unstructured meshes require more computer memory and computer time.
While they are more suitable for complex geometries, the structured meshes are
efficient to use for simpler geometries.
The generation of structured meshes or grids will be considered in Chapter 9.
This section considers the structured grid shown in Fig. 5.2a and describes the
numerical solution of Eq. (2.2.24) with the finite-volume approach. The purpose
of the indices i, j here is different from the purpose of indices used in the finite-
difference methods. There they refer to a series of discrete grid points, while
indices in Fig. 5.2a serve to identify specific cells and do not coincide with any
fixed points in space. The points A, B, C and D, on the other hand, represent
fixed points in space, specifying the location of the vertices of the cell denoted
by (i,j).