Page 99 - Introduction to Computational Fluid Dynamics

P1: IWV
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to the equation 2D BOUNDARY LAYERS
∂ ∂
r ˙ m − r
= 0. (4.32)
∂y ∂y
Then, it follows that

/2) − 1
d
d

exp (P c n
n = + − d , (4.33)
P N P
exp (P c n ) − 1

/2) − 1
d
d

exp (P c s
s = + − d S , (4.34)
S
P
exp (P c s ) − 1
where, the cell Peclet numbers are evaluated using the harmonic mean (see Equa-
tion 2.58):

˙ m n y n y n − y P y N − y n
= + , (4.35)
P c n = ˙ m n

n
P
N

˙ m s y s y s − y S y P − y s
= + . (4.36)
P c s = ˙ m n

s
S
P
Thus, substituting Equations 4.33–4.36 in Equation 4.30 and combining the
latter with Equations 4.27 and 4.29, we can show that the discretised version of
Equation 4.26 takes the following form:
d
d
u
d
AP = AN + AS + AU + S V, (4.37)
P
N
P
S
where
r n ˙ m n x
AN = , (4.38)
− 1
exp P c n
r s ˙ m s x exp P c s
AS = , (4.39)
− 1
exp P c s
u
AU = ψ ω, AP = AU + AN + AS. (4.40)
EI
In deriving the AP coefficient, use is made of the mass conservation equation. Thus,
n ∂ n ∂
(ρ ru)dy =− (ρ r v)dy
∂x ∂y
s s
=−(r n ˙ m n − r s ˙ m s ) (4.41)
∂ n ∂ψ
= dy
∂x s ∂y
ω
d u
= ψ − ψ EI . (4.42)
EI
x
Finally, the node-indexed version of Equation 4.37 can be written as
u
AP j j = AN j j+1 + AS j j−1 + AU j + S j V j (4.43)
j