Page 284 - Introduction to Computational Fluid Dynamics

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9.3 METHOD OF TWO LINES
In writing these equations, it is again assumed that Equation 9.7 holds. Now let 15:41 263
AE i, j AE i, j+1
∗
ae i = , ae = ,
i
AP i, j AP i, j+1
AW i, j AW i, j+1
∗
aw i = , aw = ,
i
AP i, j AP i, j+1
AN i, j AN i, j+1
∗
an i = , an = ,
i
AP i, j AP i, j+1
AS i, j AS i, j+1
∗
as i = , as = ,
i
AP i, j AP i, j+1
Su i, j Su i, j+1
∗
d i = , d = . (9.16)
i
AP i, j AP i, j+1
Using these definitions, Equations 9.14 and 9.15 can be written as
∗ ∗ = ae i ∗ ∗ + aw i ∗ ∗ + an i i, j + b i , (9.17)
∗
i, j i+1, j i−1, j
∗
∗
∗
∗
i, j = ae i+1, j + aw i−1, j + as ∗ ∗ + b , (9.18)
∗
∗
∗
i i i i, j i
where
j = j + 1, (9.19)
∗
b i = as i i, j−1 + d i , (9.20)
∗
∗
∗
b = an i, j+2 + d . (9.21)
i
i
i
∗
Equations 9.17 and 9.18 represent two equations with suffix j . Our interest is
to solve them simultaneously. To do this, let
∗
∗
∗
∗
= A ∗ + B i+1 + C , (9.22)
i i i+1 i i
i = A i i+1 + B i ∗ + C i , (9.23)
i+1
where suffix j is dropped for convenience. We now evaluate ∗ from Equa-
∗
i−1
tion 9.22 and substitute this into Equation 9.17. After some algebra, it can be
shown that
= α 1i ∗ i+1 + α 2i i + α 3i , (9.24)
∗
i
where
ae i
α 1i = ,
1 − aw i A ∗
i−1
aw i B ∗ i−1 + an i
α 2i = ,
1 − aw i A ∗ i−1
aw i C ∗ i−1 + b i
α 3i = . (9.25)
1 − aw i A ∗
i−1