Page 238 - Computational Fluid Dynamics for Engineers

7.4 Computer Program BLP 227

The elements of the Aj matrices are calculated from Eq. (4.4.32a). Using the
definitions of Aj, fj and Cj-i, we find from Eq. (4.4.32c) that for j = 1, , . . . , J,
2

( a n ) j = 1 (au)j = -'-% ~ (713)j (ai 3 )j = if(7i3)i
2
(«2l)i = (S3)j ("22)j = («5)j " (723)j (023)j = ( l ) i + y (723)j
s
(7.4.13a)
To find the elements of the 7^ matrices, we use Eq. (4.4.32b). With Aj
defined by Eq. (4.4.32c) and Bj by Eq. (4.4.31b), it follows that for l<j<J,

a
(7n)j = |("23)j-i + y I ( y j t " 2 1 ^ - 1 " ( 2 2 ) i - i | |
H +
(7l2)j hn) Ol2)j-r (^13)j-

(7l3)j = [(7ll)j(«13)j-l + ( 7 l 2 ) ( ^ 2 3 ) j - l ] / y

(72l) j = { ( S 2 ) j ( a 2 l ) j - l - ( s 4 ) j ( a 2 3 ) j - l

5
+ y [ ( 4 ) j ( a 2 2 ) j - i - ( 5 6 ) j ( a 2 i ) j - i ] | ^ 0
fto
5
, v~wj
{122)3 {( 6)^-(^2)j + (721^
2
(7.4.13b)
(723)j = (72l)j(«12)j-l + (722)j(a22)j-l - («6)j
^ 0 = {<Xlz)j-l(0L2l)j-l - (<X23)j-l(<Xll)j-l
hi
[ ( a i 2 ) j - l ( « 2 l ) j - l - ( « 2 2 ) j - l ( « l l ) i - l ]
hj
Ai = ( « 2 2 ) j - i y - («23)j-i
To summarize the calculation of j and Z\j matrices, we first calculate a ^
/
from Eq. (7.4.11b) for j = 0, 7 ^ from Eq. (7.4.12b) for j = 1, a ^ from Eq.
(7.4.13a) for j = 1, then <y ik from Eq. (7.4.13b) for j = 2, a ik from Eq. (7.4.13a)
for j = 2, then 7 ^ from Eq. (7.4.13b) for j = 3, etc.
In the second part of the forward sweep we compute Wj from the relations
given by Eq. (4.4.33). If we denote the components of the vector Wj by

(wi)j'
Wj (w 2)j o<j<J (7.4.14)

Then it follows from Eq. (4.4.33a) that for j = 0,

(wi)o = (ri)o (^2)0 = (7-2)0 (^3)0 = (7-3)0 (7.4.15a)
and from Eq. (4.4.33b) for 1 < j < J,