Page 200 - Introduction to Computational Fluid Dynamics

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6.3 UNSTRUCTURED MESHES
We now recall that 0 521 85326 5 May 25, 2005 11:10 179
∂
q i = ρ u i −
, i = 1, 2. (6.58)
∂x i
Therefore,
2 2
∂
i i

(q n A c ) k = ρ ck ck β u i ck −
ck β 1 . (6.59)
1
i=1 i=1 ∂x i ck
Now, for brevity, we introduce following notation:
2
i

C ck = ρ ck β u i (cell-face mass flow) (6.60)
1 ck
i=1
and
2
∂
∂ i
−
ck A ck =−
ck β 1 (normal diffusion). (6.61)
∂n ∂x i
ck i=1 ck
Thus, the total transport across the kth cell face is given by

∂
q . (6.62)
( · A) ck = C ck ck −
ck A ck
∂n
ck
Note that the normal diffusion is evaluated directly in terms of a normal gradient
rather than in terms of resolved components in ξ 1 and ξ 2 directions as was done
on curvilinear grids (see Equations 6.30 and 6.31). It is this feature that makes our
diffusion transport evaluation equally applicable to 3D polyhedra.
The convective and diffusive contributions to total transport across each cell
face k must now be evaluated. In the literature [19, 46, 20], these contributions
are evaluated in a variety of ways, but without invoking any line structure. The
approach adopted here recognises the importance of a line structure analogous to
the one available at the cell face of a structured grid. The existence of such a line
structure at the cell face of an unstructured grid, however, is not obvious because the
line joining cell centroids P and E intersects cell face ab in an arbitrary manner, as
shown in Figure 6.10. Therefore, a line structure must be deliberately constructed.
This matter is considered in the next subsection.

6.3.3 Construction of a Line Structure
Our interest is to evaluate total transport (Equation 6.62) normal to the kth cell face.
To carry out this evaluation, consider the more general face construction shown in
Figure 6.10(b). This figure is again drawn more elaborately in Figure 6.11 to carry
out the necessary construction of a line structure.
The construction begins by drawing two normals (shown by dotted lines) to ab
passing through e and c. Now, two lines parallel to ab are drawn passing through
nodes P and E. Let the line through P intersect the face normal through e at P 1 and