Page 191 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – COMPLEX DOMAINS
In these expressions, A is given by the convection scheme employed (see Chapter
5) and source D is given by
o
r ρ J o
D = r P J P S + P
t P
+ AC e ( ne − se ) + AC w ( sw − nw )
+ AC n ( ne − nw ) + AC s ( sw − se ). (6.38)
6.2.5 Pressure-Correction Equation
The appropriate total pressure-correction equation in Cartesian coordinates has
already been derived in Chapter 5 (see Equations 5.57 with boundary condition
5.58). Transforming this equation to curvilinear coordinates, we obtain 2

2
2
∂ ρ r α dA ∂p ∂ ρ r α dA ∂p
1 2
+
AP uf1 AP uf2
∂ξ 1 ∂ξ 1 ∂ξ 2 ∂ξ 2

l
l
∂(ρ) ρ rU 1 ρ rU 2
= rJ + + . (6.39)
∂t ∂ξ 1 ∂ξ 2
When Equation 6.39 is solved, the p distribution is obtained. The next task is to

recover the mass-conserving pressure correction p = p − p . To evaluate p ,

sm
sm
m
we need to calculate p = 0.5(p + p ) from solution of Equations 5.111 and
x 1 x 2
5.112. Thus, to calculate p , for example, we write
x 1
2 l 1 1 l 1 1 l
1
1
2
1
∂ p ∂ β β ∂p β β ∂p
= +
∂x 2 ∂ξ 1 J ∂ξ 1 J ∂ξ 2
1 P P
1
1
1
1
∂ β β ∂p l β β ∂p l
2
2
2
1
+ + = 0. (6.40)
∂ξ 2 J ∂ξ 1 J ∂ξ 2
P
With reference to Figure 6.4, the discretised version of Equation 6.40 reads as
1 1 1 1
1 1 p − p l + 1 2 p − p l
l
l
β β
β β
J e E P J e ne se
1 1 1 1
1 l l 1 l l
β β
β β
− 1 p − p W − 2 p nw − p sw
P
J J
w w
1 1 1 1
1 l l 2 l l
β β
β β
+ 2 p − p nw + 2 p − p P
N
ne
J J
n e
1 1 1 1
1 l l 2 l l
β β
β β
− 2 p − p sw − 2 p − p S = 0. (6.41)
se
P
J J
s s
2 In Equation 6.39, cross-derivative terms containing dA 12 are dropped. This is because the pressure-

correction equation is essentially an estimator of p and, therefore, in an iterative procedure the
m
truncated form presented in Equation 6.39 suffices. It is of course possible to recover the effect of
the neglected term in a predictor–corrector fashion. U are contravariant mean velocities.