Page 198 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0 521 85326 5
May 25, 2005
0521853265c06
6.3 UNSTRUCTURED MESHES
The lines joining vertices will henceforth be called control volumes or cell faces 11:10 177
and elements will be referred to as cells. Thus, a triangular element will have three
cell faces. The same logic extends to polygonal cells. Now, it is easy to recognize that
when nodes are defined at the centroids of cells, there is no node at the boundary
to facilitate implementation of the boundary conditions. Therefore, a boundary
node must be defined. We adopt the convention that the boundary node shall be at
the center of the cell face coinciding with the domain boundary. This is shown in
Figure 6.9(b) by point B. It will be recognised that even if there is a change in
boundary condition on either side of a vertex, the boundary condition can now be
effected without any ambiguity.
Practitioners of CFD familiar with control-volume discretisation on structured
grids prefer the element-centred approach [5, 46, 20] rather than the vertex-centred
approach. In the discussion to follow, therefore, the element-centred approach is
further developed.

6.3.2 Gauss’s Divergence Theorem

The transport equation (5.1) in Cartesian coordinates is again considered here but
3
without the presence of r for brevity. The equation is rewritten as
∂(ρ ) ∂ q i ∂(ρ )
q
+ = + div( ) = S, (6.49)
∂t ∂x i ∂t

q

where the vector = iq 1 + jq 2 and i and j are unit vectors along Cartesian coor-
dinates x 1 and x 2 , respectively.
To implement the IOCV method, Equation 6.49 is now integrated over the
elemental control volume shown in Figure 6.9. Thus, with the usual approximations,
we have
V

o o
q
ρ P P − ρ P + div( )dV = S V, (6.50)
P
t V
where V is the volume (i.e., the area in the 2D domain with unit dimension in the
x 3 direction) of the cell surrounding P. This cell volume can be calculated knowing
the coordinates of the vertices.
The second term on the left-hand side will now be evaluated by invoking Gauss’s
divergence theorem [70] applicable to a singly connected region. Thus,

q

div( )dV = q · A, (6.51)
V C

where is a line integral along the bounding surfaces (or lines in two dimensions)
C

of the control volume and A is the local area vector normal (pointing outwards) to
3 This neglect in no way disqualifies the developments to follow.