Page 207 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0 521 85326 5
0521853265c06
186
2D CONVECTION – COMPLEX DOMAINS
The following comments are now in order: May 25, 2005 11:10
1. Equation 6.103 has the familiar form in which the value of P is related to its
neighbors Ek .
2. Superscripts l and l + 1 are now added to indicate that terms D k containing
Cartesian derivatives of are treated as sources and therefore lag behind by
one iteration. The same applies to the source term S. A method for evaluating
nodal Cartesian derivatives is developed in the next subsection.
3. Equation 6.103 applies to an interior node. When the control volume adjoins
a boundary, one of the cell faces will coincide with the boundary. In this case,
Ek for the boundary face will take the value of B , where B is shown in Fig-
ure 6.9(b). For different types of boundaries, boundary conditions are different
for different variables. Therefore, Equation 6.103 must be appropriately mod-
ified to take account of boundary conditions. This matter will be discussed in
Section 6.3.10.

6.3.9 Evaluation of Nodal Gradients

To evaluate the D k terms in Equation 6.103, Cartesian gradients of must be
evaluated (see Equations 6.90 to 6.93). This evaluation is carried out as follows:

1

∂ ∂ ∂
= = dV. (6.104)
V
∂x i P ∂x i V ∂x i P
P
The volume integral here can again be replaced by a line integral and subsequently
by summation. Thus
NK
1 1
∂
i
i
= β = β , (6.105)
1 c 1 ck
V V
∂x i P C k=1
where
]
ck = [ f mc E 2 + (1 − f mc ) P 2 k
= [ f mc ( E + E ) + (1 − f mc )( P + P )] . (6.106)
k
The appearance of P in Equation 6.106 suggests that Equation 6.105 is
implicit in ∂ /∂x i (see Equation 6.90). However, since the overall calculation
procedure is iterative, such implicitness is acceptable.

6.3.10 Boundary Conditions

To describe application of boundary conditions, consider a cell near a bound-
ary (Figure 6.12) with face ab coinciding with the domain boundary. Note that