Page 211 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – COMPLEX DOMAINS
6.3.11 Pressure-Correction Equation May 25, 2005 11:10
In Chapter 5, the total pressure-correction equation in Cartesian coordinates was
derived to read as

l

∂ p ∂p ∂ ρ u i ∂ρ

= + , (6.128)
i
∂x i ∂x i ∂x i ∂t
where
ρα V
p

= . (6.129)
i
AP u i
p

In this definition of
, α and AP are, respectively, the underrelaxation factor and
u i
the AP coefficient used in the momentum equations. Invoking the Gauss theorem
again, the discretised version of Equation 6.128 will read as
NK NK NK
V

o p

AP p = AE k p Ek − C ck − ρ P − ρ P + D , (6.130)
P
k
k=1 k=1 t k=1
NK
where AP = AE k and
k=1
p
p (
A) ck
AE k = d = . (6.131)
ck
l P 2 E 2
Two comments are now important:
p
1. The D term in Equation 6.130 will contain Cartesian gradients of p . However,

k
duringiterativecalculation,sincethepressure-correctionequationistreatedonly
p
as an estimator of p , D is set to zero.

k
p
2. Evaluation of
ck in Equation 6.131 will require evaluation of V and AP u i
at the cell face (see Equation 6.129). The evaluation of cell-face volume can
be accomplished via a fresh construction at the cell face as shown in Fig-
ure 6.13. The construction involves drawing lines parallel to ab passing through
P 2 and E 2 . Then, two lines parallel to normal n (and, hence, parallel to line
P 2 E 2 ) are drawn through a and b. The resulting rectangle c 1 –c 2 –c 3 –c 4 will have
volume
. (6.132)
V ck = l ab × l P 2 E 2 × 1 = A ck l P 2 E 2
Using this equation therefore gives
2
α (ρ A ) ck
AE k = u , (6.133)
AP ck
u
where AP = AP u 1 = AP u 2 can be evaluated from formula (6.66). 6
ck ck ck
6 Alternatively, one may evaluate AP exactly by carrying out a structured-grid-like discretisation
u
ck
over the control volume c 1 –c 2 –c 3 –c 4 . This is left as an exercise.