Page 205 - Introduction to Computational Fluid Dynamics

P1: IWV
May 25, 2005
11:10
CB908/Date
0521853265c06
184
where 0 521 85326 5 2D CONVECTION – COMPLEX DOMAINS
l P 1 e i
l xi = x i,e − x i,P − β 1 (6.87)
A c
1
d xi = x i,c − x i,e = (x ia + x ib ) − x ie , (6.88)
2
(
2

i A c . (6.89)
1
l P 1 e = l Pe · n = (x i,e − x i,P )β

i=1
Now, since coordinates of e, P, a, and b are known, using Equations 6.86 and
6.80, we can write Equation 6.77 as
2

∂
= P + P = P + (l xi + d xi ) . (6.90)
P 2

∂x i P
i=1
Invoking similar arguments, it can be shown that
2
(1 − f m,c ) ∂
= E + E = E + d xi − . (6.91)
E 2 l xi

f m,c ∂x i E
i=1
Now, a and b are evaluated as the average of two estimates in the following
manner:

a = 0.5 P + l Pa ∇ P + E + l Ea ∇ E , (6.92)

b = 0.5 P + l Pb ∇ P + E + l Eb ∇ E . (6.93)
6.3.8 Final Discretised Equation
Substituting Equations 6.90 to 6.93 in Equation 6.76 and performing some algebra,
we can write the resulting discretised equation as
NK
V

o o
ρ P P − ρ + C ck [ f c P + (1 − f c ) E ]
P P k
t
k=1
NK

− d ck ( E − P ) k
k=1
NK

= S V + D k , (6.94)
k=1
where

]
− 0.5( a + b ) + (1 − f m,c ) P 2 k
D k =−d ck B ck [ f m,c E 2
+ d ck ( E − P ) k − C ck [ f c P + (1 − f c ) E ] . (6.95)
k