Page 209 - Introduction to Computational Fluid Dynamics

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It is easy to show that 2D CONVECTION – COMPLEX DOMAINS
l P 2 B i
− x i,P = l xi = x i,B − x i,P − β , (6.113)
x i,P 2 1
A B
(
2

i A B . (6.114)

1
(x i,B − x i,P )β
l P 2 B =
i=1

Thus, Equation 6.108 can be written as
q

( · A) B = C B [ f B ( P + P ) + (1 − f B ) B ] − d B [ B − P − P ] ,
(6.115)
where the diffusion coefficient is given by

B A B
d B = . (6.116)
l P 2 B
Using Equation 6.115, implementation of boundary conditions for scalar and
vector variables will be discussed separately.
Scalar Variables: For the near-boundary cell, Equation 6.103 is first rewritten
as
NK−B NK−B
V V
ρ P o + AE k l+1 = AE k l+1 + ρ P o o P
Ek
P
t t
k=1 k=1
NK−B

l
+ S V + q (6.117)
D − ( · A) B
k
k=1
where NK − B implies that the boundary face contribution is excluded from the

q
summation and accounted for through the −( · A) B term. This accounting can
now also be done via Su and Sp as

Su − Sp P =−( q · A) B
=−C B [ f B ( P + P ) + (1 − f B ) B ]
+ d B [ B − P − P ] . (6.118)
Thus, when B is specified, it is possible to write
Su =−C B [ f B P + (1 − f B ) B ] + d B [ B − P ] ,

Sp = C B f B + d B . (6.119)

Sometimes, boundary influx F B =
B ∂ /∂n | B is specified. Then, it can be
shown that

Su =−C B [ f B P + (1 − f B ) B ] + F B A B ,
Sp = C B f B . (6.120)